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Math for Economists - Lecture 6

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Again ready to Rego who started Chapter 10 today for The way things are broken down the 1st 2 Chapter 7 and 8 were basically about solving systems of equations well actually 7 was about systems of equations and 8 was about as major sees a little bit And then 9 we can we we continued the theme of studying matrices but then can we tied up and we said that it all up With Kramer's role and that kind of combine everything the nature's matrix theory with The systems of equations in Chapter 10 it's a little bit more abstract and it's a little bit harder Chapter 10 It is hard for you guys because this is where you really get Challenged on year mathematical maturity just because we have a lot of abstract ideas and that if you're not used to that you're dealing with that while you're trying learn the map of the combination makes it difficult to give yourself a break Pegasus with some users Chad I is much as you can and if you look in Aden it's just to deeper too difficult for you to deal with the amount of time it took a letter that some these things go territory is really clear on exactly what types of questions on the output from this on Inter as we go out and say OK this this example is something we need to know so that what is what's gonna happen and really all losses in the abstract Shannon United know what's going to be tested nuclear on 1 of the things that we need to do it this is just to develop a little bit of an intuition that title of the chapter advanced topics and linear algebra that's intimidating by itself does that language had advanced years math class see name it's already a little bit intimidating and then what about linear algebra we've been studying linear algebra without really using that term the way 1 of think of linear algebra is it's really just the extension of the Situation you had a nice school and high school you have the equation 2 x equals 3 then linear algebra is a generalization of that if you want to give it that way where instead of having to x equals Syria now have 2 variables so you have a tricks times the 2 variables equals the number 5 So it's like this it's a 2 x equals 3 that's the simple case put that linear algebra that aligned a linear equation used solid using algebra those terms that you learn when you were younger so the return when you're out a reprise of the stuff that you were doing when you're younger sister was a necessary introduced that fancy language in a match for me have something like 2 Rex equals 3 and X minus while equals 0 and the gave us an equation for we have 2 1 1 negative 1 X Y equals 3 0 Here we solve this by saying x equals 2 Wyndhurst 3 times 3 you like 3 out here we solve this Saying that X Y was equal to overtake the inverse of this matrix here the of this nature But at you 1 1 negative 1 in numerous times 3 0 and that gives us a solution on 1 that's linear algebra even know you were doing it before this is the extent And then of course about 1 of 3 variables omega-3 3 by 3 matrix here for variables of continues on this term is talking about this but we're not talking about as many variables as you want that what makes it abstract in hand The complicated some types If sex in Section 10 . 1 that is of them This is the vectors That's another abstract terms that's another story look at that and say what is that great go against some mathematical stuff will just take it apart a 1st of all hardly ever seen the term vector vector yet sector was a special make major cities that had just 1 row or 1 call those were vectors to call the roll that er or column vectors case we had this term in your mind you just wanna thinking think of something By column or row column vector for and is wrote that both sectors again now that the word space for will will be a vector space Willard the words states implies that this is a space accumulating in this space this room a vector spaces where all the sectors are located focus of typical puts have some typical examples here examples of vector space was just so you have something that I have in your mind when you see that term the 1st example is 1 that you're very familiar with the real number is that the vector space but what I'd like to do is I'd like to draw see you see that it's a vector spaces sectors led a year 0 call that the origin And in what is a vector look like in this vector space you have something like makes the number X that 2 or something like that that the sector right here that the number acts that Wang by Apple narrowed to it that's called the vector and that's a vector space the vectors emanate from the origin and then go out and connect dealt with the location of the war that's a vectors another vector spaces where are the which are to be be favors are times are or or by are our class of those rocketed interchangeable terms orders are to look like but that 2 copies of R have Happy here where the X coordinates and the Wye accord both of those are copies of Are you could think of this as are cross car that's how we get to work At the start and that are ladies me and I can see points in this two-dimensional space vectors what would a vector look-alike venture would be something like on Wednesday was your X and Y OK so what is that that I go out the blank X and I go up this My wife just like a regular quarters that you used to have I gave the . 2 3 you would not go to the extraction and 3 in the wider action the only difference the you're used to just putting a point there Now we're doing in the vector spaces weakened at that point To the origin and this is a right angle here that are too that's a vector space 3rd 1 Orcas see if you can just forget what the 3rd one's going and if this is the 1st time that a one-dimensional vector space that I go to dementia later 2 copies well then goats are 3 this is to be three-dimensional but really it's 3 copies of Are you don't wanna right it that way This is cumbersome What is it look like what time that I have 3 coordinates but again the origin is important it's the center point of whole the sectors in order to emanated from the origin out to the point here they've emanated from the origin of the point it will be no different here I've got ahead of vector Kennedy gives a perspective that could lying down and see where it's at the point that it's above this is your that lets me and then gotta have water in 3 dimensions and X Y and Z
So he is equal to the court X Y Z that sectors 3 messages use the language used to talking about these things in with different language had to be careful about that Hedges said 3 space that sort of a sure short way of saying art 3 or a vector space in 3 dimensions So just examples of a vector space is a is a space where you have a bunch of actors drawn all of that As possible they could begin go all over the place but I just Juergen Eric 1 which sort of represents all on the same time same thing with this sector's go all over the place even negative will you do with the vector space who would we want that do a couple of things
1st of all we're China find what is the length of a sector like 1 is you can start to imagine how do we find the light
That's something will hear it's easy the length is just going be the number pad number to the length will be too if I have the number negative to it'll still be too because it's the the distance from 0 to be the distance with that from now I can't say 0 because that's specific to the real numbers would I wanna talk about the 0 here and 0 here and 0 is there a need to use the word origin that way encompassing all all possibilities from the Orange will come down the Pythagorean theorem lives check check out or 2 They are But The real numbers I just that in their mostly warranted immediately an hour to an hour 380 even are for some always going go backed are examples Pocket was look at a vector in a bar But find the light of it Now this is just a generic ones so here's here's my vector and then it's got coordinates X so that means this distances X and this distance here's why So how I find this link here The link there the Pythagorean theorem right isn't that the length is equal to the square root of 6 square sliced it sisterlike squared is eager let square was why square that's the Pythagorean colors years agency squared equals X. propose y squared Add to that the formula for the life but really satisfied and after we have arranged things out blank has some symbolism to represent that of the symbol that we use here It's like absolute value but it's the double line that the notation for length double line of that's going to be that way and said here that the notation for double I put double lines around anything it's gotta be a vector it's gotta be something that just has to use 1 column or 1 row and whatever's in that row X and Y and religious square that square that
Adam take the group if so that's two-dimensional Chris world better go to 3 dimensions see what's on America let's get a tap the skater a random vectors here
But to be persuaded that is Here is a look at quarter of the year by the unit X Y and Z illustrated drug in a little bit and see what's going on here so I draw the height here that Ziegler that's the height of the vector Ohio idea That's a right angle there and then we can draw and baby girl A little of that triangles down the there so how I get to this point right here this is really equals Is true X y z so means that the x coordinate system access disease so what's this distance here this is why this little distance years X and so what that means is This piece rate Theories see how that the hypotenuse of that little triangle on the basis of this He's staying here is the square root of square The ballots so what I've got here let's But sank off after yellow triangle and it over here the yellow triangle is me years The length of TV sets the length of each then down here I've got square root events were plus Y squared and then over here that the Vertical component Z so as just another right triangle with the Pythagorean theorem this case V Back to
OK highway D had 0 I get this it's going to be the square root of that squared plus ceased to just that site was that said
Squared and then the squared away here diskette were close square which is great that of the new notice Your notice that if I go from 2 dimensions to 3 dimensions I basically have the same formula I just take the coordinates of the vector images square foot
Now I get all them out of the way to do in general on them do a few examples to you can see from from those 2 examples that there's nothing has really nothing special about the dimensions of of the Vector space if I have 3 dimensions the wave that just put all 3 quarters 2 dimensions is too so it makes sense by analogy that if I have an end dimensional end copies of ah then a van What is that journey look like in
This space or that have ended accord that I wasn't are 3 3 different Hornets into it to form makes sense that I have an of them so high A label that label attacks on tax and that ended coordinates and then length just by analogy what should it be that should be X 1 square plus sex to square all again under X and squared you don't have to rate to but in general focuses is easy computation on the exam also years of vectors go find meal like that we'll just have to know this formula and computed This is the general case this is your final formula for all possible vectors analysis couple examples of easy the reform or battered even need to tell you what vector space it's in that just vector has 2 components so it's living in or 2 but the likes of it is just computed as square root of 3 summers Square which by We just continue on here pilots at 1 2 negative that not to tell you what spaces is just a vector here's a vector go find the likes it after drawing the graph theory can be deceptive Field If you but just as take each of the component square Adam up Dick squarish 1 square 2 square plus negative Jews square make square the negatives there shouldn't be any negatives when you have these things that because when you square negative becomes positive article wary of 9 which is 3 that's the life of that sector if
Now I have received very have all possibilities so I give you a vector that say has 5 comport with the example of this
He wants you know How do this there's nothing I can do to trip you up you know the formula the formula if I want to look like that Then I take the square root and square each her to square Negative cues where 3 squared plus 2 where last nite to maintain DDU 50th does 50 I wouldn't do that because I do not have a calculator and press that he didn't get the hood 50 5 is a good number so that's what we here at 16 the length of that sectors also fired in Moscow his numbers and it still has a length of 5 to 1 think of that is that is just an arrow see you can't picture 5 dimensional space so will you do with your mind when you see this yet there's something for your mind to do just use this picture not three-dimensional that's all you can do so when I see that I spoke got a better 1 note that like this and it it was 3 years Newark scribbles wife group of these square but I've got 5 others do the same thing 5 OK that's a length of vector 3 simple thing we gotta have that ready for losing by the next issue is that Edition this is part of what makes it a vectors space that words space is I can I can take things in it just like in the real numbers could take any 2 real numbers and combine amendment on I get another real number has this property worth by combining 2 things it stays in that world that that words space that's that's more advanced way of looking at it so vector edition was just do this by example here I have a vector you will do this in 2 dimensions just use It's very obvious what you're gonna do that Especially wants Eliminating a little bit more so these vectors OK simultaneously 1 a column of vector but on the other hand just matrices right there just it's just a box of numbers this happens to be 2 by 1 misses to buy 1 matrix still to be consistent and mathematics which should be the case that I had these any differently than I would add matrices disease or majors so it's just the obvious that group plus hazardous matrix addition to 3 plus 2 0 4 But what's important is to see the picture that goes with that Vector edition has But picture merits just done abstractly with algebra using the the matrix theory but If we grab these guys to focus 0 2 0 2 0 and that's the ex Coronel Cornet so here's to 0 That's V and the other 1 is 2 3 2 3 there's the . 2 3 but then I connected with a narrow there's you so then what is you plus a look like life is calculated you plus VAT is for Commerce But the sea Yet so OK so there's 3 4 and 3 years there Scene-stealer published today years that here's the addition here That's You Let's not very enlightening book What a fight to seen this picture about to draw already is the parallel agree that you would be create some of those sectors as the diagonal GKO anyone tried a key That he keeps a visual stuff as much as you can in your mind while you're doing the algebra so the idea of adding vectors is viewed as the 2 vectors create a parallelogram and the addition is always the bag that peril That would facilitate that they're so it is algebraical 1st to see how you can calculate it without knowing the picture but to stab tuition the goes with the picture I will just like what we had Edition of matrices we also had what's called scalar multiplication of matrices but now we have risk attacks that were scalar multiplication to these vectors Pamela tell you right now fears knows with scalar multiplication does it All it does is it changes by multiplied by a scalar than that it's like a scale model and multiple a reeling by the same number shrinks everything by the same amount of but by 1 after the shrinks back multiplied by 10 am BILLING grows by skilled and just matrices so if I have 2 3 but the column vector a matrix wherever were you want use here in a vector space it appropriate to use the word vector who what the scalar multiplication let's committee something like 2 Multiplied by a scalar And what that to Time to 3 which gives you for 6 Mallet see what have I had the sector to the list of space here There's that there's the vector Chiu X quarter to 3 is here that and then if I do to me that's for 6 see it is doubled each component And what that It's skills here there's important skill as a couple for scalar stay you wanna pay attention to so this is The state straightforward but if I were they have the same negative but see what happens with that is is just so What you want intuition what happens if I happen multiply sector By negative what happens visually OK so I can write it down algebraically that's negative times which is now negative negative and biographical here's V that's V and then minus V will be minus 2 minus 3 so there's minus see it flips it around the origin of fire multiplied And that's also true in the real numbers I have a 2 year multiplied by a negative thoughts of overt negative Of this sector here there's V Your mind is also true in just a regular role real numbers but would extend a here then Becomes all the more confusing what happens to that sector would multiply that negative wanna think of this flipping it over the origin
The shift of craft But Next concept is a linear combination of that even know what look at this stage from them
What will I put on your test It was 1 key factor that will through the section that's that like the sector so far this is just a introducing these ideas will build on this then will get questions that will be on the exams so far it's just the like the vectors on this concept of linear combinations will be will be there but that you get to the problems while Manila Stewart a linear combination is through an example but think about the language ear The combination of vectors on combining sectors and that Clairol said that more than 1 vector facility star up with you as someone warned and 2 0 so there's 2 vectors nominee do a linear combination of vectors I'll tell you what it is a linear combination its I combined the 2 sectors but what I allow for as I might multiply each won by a scale So it's just like
Edition of vectors except might Under combine these 2 issues there's the scalar multiplication multiplied sector by a number and then there's edition
By combine the 2 ideas that's what you get better to call the company combining the 2 so linear combination of you and V is written a Times we must be times we just means I comprised of allowing Dabur gala of the school on a scalar about 1 and then I'm gonna have to think of adding as being the diagonal the parallelogram will see this picture case you 1 1 there's you And then 2 0 at 2 0 so there's V 2nd the yellow But There's a new pay now on manager the linear combination to the EU plus 3 v I can ride out what that is it's 2 times this but We can also raise here what to you To you is I traded up to you is here and then please There and solidify add them Then I get this diagonal of parallel to you you could say with the coordinates are multiply 2 times you that gives me to to let this guy and then 3 times V-neck takes me all the way up to 60 cent at 6 On the matter I get that back peril to really what I want you to think of it is if you have a linear combination of vectors it's you have your 2 vectors I might multiply and buy something and extend the length of those that parallel the 1 awry added
I'm just going get that no that's what makes it really complicated is that we can do this in 3 dimensions and with many many Vector 0 that just 2 that's just the simplest case This is the real concept were headed toward the near independent and with independence comes deep
The 2 worlds here start this big concert who wanted to get a handle on so that start out is seen in our minds what this some examples of independent Q. a R 2 is 2 dimensions I got The victory here The court taxis X and Y and if I have you and say We those are independent told you exactly what it is of the same sets an example of the independent directors linearly The thing about the language why would we attach toward linear to this independents Well near Implies line of fire Wyoming when you're in 2 dimensions and when you get to the 3 dimensions it implies either liner a plane It's not always just wanted here So that's independent so what is deeper and deeper and ended his that the vectors are on the same line that linear deep and nearly 2 dimensions to vectors being on the same line here's the lines Have you here and I might have to be in the opposite direction the user dependent our goal is to learn how to detect this phenomenon if I give you a couple of vectors are due to 3 vector sum I give you a pile of vectors and I say are they independent Journal urinal wanna test for the train our just finding out what it is if he a huge
1st Let's see what it is and 3 dimensions because it's more complicated 3 mentions if I have a 3 vector so it's a little extra X zombies
And while I got 3 dimensions if I have 3 sectors All going in that looks like it's This direction so Carlos Ike 3 sectors on different directions not even in the same plane is in the targeted draws you deserve now here deepened in in the three-dimensional case 2 different versions of independent So this is what I was saying a 2nd when you have 2 vectors in 2 dimensions their dependent on the same line but in
In 3 dimensions you could have this you could have 1 Swaney here and all the vectors are in that place this is just so that's the analogy analogous situation of this 1 but now retired dimension if they're all the same plane that's also considered dependent and then you could also have that Casey
In 3 dimensions wrong same line which make it OK so that a throng the same line The have said You and the Maggie W those all on the same lines as it deep but sort of worse than this year they have some freedom here they're all the the same plane a year they're all walked in on the same line that sort worst form of dependents
Hey if it so if 9 EDT the actual mathematical that conditions which is an accomplished offices just worry realistic deal abstract but will get through it
Look at here's the mathematical definition those are the ideas is the pictures that come with this concept but in order to work with it we need to have some mathematical that Definition so we can check things algebraically because we can always see what's going on inside dimensions of the to have some way algebraically to detect this phenomenon without having to drop pictures Ocatillo will just starting are you here 2 vectors are home to be independent status satisfy this condition Let's have a duty you're not used to mathematical statements that they have to work a little extra hard here to decipher what's going on here 1st symbols Lander 1
They say this in English the Greek letter Lambda sub outlander won 1st is a constant What is this is can use it we can attach our language that we've learned this is just a linear combination of U.N And what it's saying is it's equal to 0 so imagine what the 0 victory and that the The origin of 0 0 rate that's it for all the components of 0 so what this is saying is that they only linear combination of these 2 sectors that gives me 0 is the 1 where I picked these constants to be 0 the 1st OK So what this this is just puerile diverted to show the an example here would show But that he really is gone negative ones him and he is 3 0 show that those 2 guys are in and I can't abbreviated a linear Independence until let me say some writing I would lead to war But here where the directed that you don't to read Sewanee use this definition to show that these 2 sectors are in this is the kind of thing you yet I knew that may not be that simple use 3 vectors but would hold a little bit more your this award real good test for some of the things that makes a right now orders use the definition and see if we can handle that OK where's it said it says that if you had that situation that must mean that land and landed 2 or 0 Casillas give ourselves a situation Manitoba 1 a new plan to meet their Ellis suppose have planned no 1 knew it meant to be
Equal 0 The Muscat picture where we have a picture for this can intuitively see what's going on here is 1 negative 1 And 3 0 We you what to all linear combinations look like they all look like the diagonals Of parallel for do as I said modify this 1 shrink it Now that more than I had it said 2 times I 1 no matter what I do I'm always apparel appeared to see that no matter how These things I'm still a parallelogram and the diagonal alarm is not easy because it'll have apparel now the only way to get back to 0 is by multiply this by 0 . 0 then went out it that's the idea Of and also I could see from the picture they are independent they are they're not on the same line so I can only say that their pocketbooks get back to this so what does this say this Planned 1 times with 1 negative point plus plan 2 times 3 0 equals 0 0 when I say 0 in the sector space If scattered be so that's a little bit of a you think that this is maybe you know like numbers equal 0 when I say 0 here vector space so that the 1st step is to realize that 0 in the vector spaces all components And what is this said this is the same as saying how well I can put a altogether that landed wanted plus 3 landed too That's on the top and the bottom I have slammed 1 and that's all that it be 0 0 just matrix multiplication if I happen more By that out eg Atlanta 1 negatively No 1 of the biggest 3 the 2 men add the matrices They get back OK so what that is is this is a system of equations this implies that way 1 3 landed 2 equal 0 and minus slammed 1 0 And a memorable is
We assume we have this statement and we wanna find out that when the woman Linda to ripple 0 or about the find that out this thing Yeah says Lander 1 0 the
That back in appear have landed 1 plus 3 of them de Chile 0 that implies 3 landed 2 0 which means when she was so look I got the result but I want this implies and want to see I found that result that so then If that's the definition of Lanier independence than I can say you see or hear That's that's at paying the UC Woodward will What we're doing were combining a bunch of things we've done before we got this new concept of a linear combination we've got the concept of adding vectors Multiplying by a scalar But then ultimately comes down to use all the system of equations Which we worry supposedly bastard have with math cause or whatever you're on your mastered it before it is not really the case but look at the way you see reusing are linear thorough study of systems of equations Solve popular wouldn't think of a short here because I know about you but Julio stuff for doing this class would like they have a short where we see in this Picture before this happened I drama picture before where I put vectors down I drew a parallel was at related to It was not related to the determinant It wasn't the case that when you had an area not equal 0 0 the the terminal was 0 And when you had area equal to 0 then you had the determine equal 0 so think about this linear independence in that context if they're on the same line they have no area so therefore the determinant of his and if they're not on the same line thinking Why could take these vectors here they choose as negative 1 1 negative 1 . 3 0 could just put those in the nature of the called and the determinant of that would tell me whether the nearly independent so there's another connection something done before The determinant is our test for linear independence in this specific case so the why what specific cases Crude Pixels Notice I have these 2 guys here would do the same example they create they give you this parallel in the area is equal to 3 so we can just do that independently but we also have this prior knowledge that if I make a matrix Well where my on EU here and I will be right there that we want negative 1 3 4 0 see there's a movie the determinants This Is equal to 3 just not equals 0 Which implies you and are Lanier defended his if they were on the same line you would get an area that's that The idea pockets illustrate this down in general so that we can do have a real problem you might get on with it owes a high-ranking combined vector space that with some of the old stuff like it is ask you determinants but I can also just do it in this context you'd be forced to do a determinant or solar system liquid and solid so that's kind of get all this codified in 1 D Are so I have and that peers in or is this is specifically these 2 match so I have that's the was 1 of those situations at 2 vectors in 2 or 3 vectors of our 3 or 4 vectors in our for years the vectors 81 pursuant to be But that's the math notation for that state vectors in Oregon These are ready at and it ended the determinant today is not 0 Panetta tell you it a S where As the Matrix where I just put these sectors Look Difference in the columns of 81 82 In not right that at an English The year so you want to do and are called on to say that this already here even know it's in the general case with and sectors and this is what you need to know from nearly to an example now that uses this information this high from where which you wanted to this is his international version of this Example right here where the vectors Are not on the same line they form an area than the terminally nonzero and I can use that to decide whether there is any of its 3 dimensions that has to be a volume of thirst for a while in there has been a four-dimensional Wally here is a gig for vectors and Marianne are for The here is not to 3 0 1 4 0 No. 4 0 0 0 These are all end our format I know they're not for his 4 components of 1 2 3 4 must be an hour for saying he did say that sorry complied But you see seeking you know whether their independent not less if we want avoid using the definition well the way we can do that is used this year end actors desire for vectors in are for Then I can just make a matrix of a DVD 1 2 3 4 his putting these guys in the columns don't have to write this step by the close of 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 walked analysis computer determinant well when I do this determined here I think I want to use this column that's the 1 that has 3 zeros and just the 1 in it so the determinant days everyone in times of determinant of the Matrix leftover when across that offers religious 3 quickly And look this is a sister remind you since have termed a week so as well as Romania some of the things we've done this is a triangular matrix So I just multiply that day I mean and all I care about is is not equal to 0 but that says it the determinant of that matrix that create from taking the vectors of putting them the columns if that's nonzero than I must admit that I had some volume which means that they had dollars was different direction so that means that he you won't be
2 3 and 4 Are the nearly and that's that panels the explanation for all the cases it you have to deal with you me need for this class this concept linear independence is a much bigger concept Batista this class
We're gonna have this situation you see the exercises most like Hey could cross by the way If you're fighting yourself you liking this kind of match that you wanna take a pattern Do you volunteer for adding that affliction that if you if you happen interior City Guide is really cool you wanna take another math class and can get really into this and so some of the classes that we offer here matter too deep and that 3 D and now 60 6 GT is required 60 does all the sets of hardcore rap class but it's for have a major computational economics quantitative that
Yet so you'll take this class if you're a quantitative And if you know that advanced you better pay attention this glass cases much more intense will take this concept will spend all week on it and do all kinds of more complicated things so you will get this basic stepped down planning on taking OK now don't even know I'm been talking a lot and I've been doing a lot of different things Dr. John pictures and stuff the things they need to be responsible for so far are the length of the vector and then I give you a pile of actors test with their linear independent by checking the determined in this specific case here where these Natcher so so far that's all you got the responsible for what she will have more get to vote of course Damiani questions for me hard
Now we're getting it to the real definition of a vector space yes This would deter deter triangular Who For triangular we before by for this was a triangular only once I got to 3 by 3 anyway that that wasn't triangular
0 0 yes Because I notice that I saw that 1 of balance triangular could urges that from now on its way yeah I didn't see that that is trying to the program OK definition of a vector space so adjusted clearer For the video and also heard her question was what we did the 4 by 4 meters could Niger's noticed that that was the triangular matrix and multiply the Daniels and the answer is yes if you multiply the 1 the negative 101 of the negative when you still get 1 as the determinant citing even in need to do this step here that she was pointing out that the thank you Exactly it's it's a triangular matrix The for the panel but anyway and see the triangle There the upper triangle Segers multiply the diagonal just like it did here except I realize that there too Just Then in the terminal B 0 you're still multiplying Elements of the bag affair happened 0 there get 0 A vector spaces 2 conditions Marty looked at the visit the 2 conditions are you can add vectors and you can do the scalar multiplication and the key that I didn't really mentioned her baby alluded to Is that when you do that you stay in the sectors so the way we write that out in mathematical symbols is a safe I have you give us give yourself sectors capital B by heavy you can be in sectors space this Little similarities in that body told you that I walls that followed 1 of few symbols used there's another 1 implies that I think you can handle that pockets of I have 2 vectors in the vector space and had friend In the vector so that's the that's the definition of a vector space 1 of them then the 2nd 1 is that I have a vector any factor in the vector space that implies that I can multiply But any scalar and its In the vector space had to tell you would see as see is as a real bacillus let's take a look at the vector spaces now and see that those regions so we can that's a chance to make a connection with a real mathematical definition and some pictures Go back to the vector space that you were are you really familiar with that young age and that's the real numbers real numbers satisfied 2 condition well of I have you and we can all are Fernandes replacing are with the there so have union our Saudi thinking just of 2 and 3 2 and 3 2 and 3 and all are Does that mean that you plus policies our yet and they reveal the co-op with any 2 examples doesn't happen because you know that we 2 numbers years real so this implies that 2 plus 3 5 and but in general yes I take any into real numbers Madam I am still very real numbers and If I take care of that And are like to and fight multiplied by a constant that in estate Cher what's said seat of that and lies that too is a our Internet implies that will see tends to his higher That's true trendy see him are that's a social definition The new dryly pictures that it helps to have pictures See going because that's also see that that's true are to real quick was checked those conditions at that you and be beauty here are What's that picture that means Dad you and me Now what happens if I do you plus All we know what that picture is now you plus is this here that's definitely in the state face assembly of the space the that's true with this sort see But picture and then what about if I had to I have 2 years in part to the that apply that CD is Well it does produce What is seen CV is just just take this and multiply by 2 of them extend its light here might be that's still in the in the sectors leave If I multiplied by native floated down here still and so what I multiplied by It's in the stadium of actors that definition I need So there for this year's ballot Exercises that you get a new book on the concept as they give you things from state is this a vector spaces you have to check these conditions The Do an example That those are vector spaces so that the conditions don't fail but when you checking something that is the 1 of the conditions must be bullets The case but look at this example is the 3rd quadrant of 388 . with the Roman numerals 3 is the 3rd quadrant of vectors schools who I know it's a 3rd third-quarter stand here will soon lower Inc. he's here I could still ask the same question about it so what is the 3rd quadrant it's laughed at all that X Y so that way Exs lessons and wisely less 0 that's I end up in the 3rd quarter as an example negative 1 negative to is in the 3rd quarter
The question is if I take 2 things in the 3rd quarter and Adam I stay in the 3rd quarter to think about that I take 2 things was drawn up picture of it if we could believe it might take something of a 3rd quadrant and take another thing 3rd quadrant and add those I know what the picture it's it's this peril So that's you know and that's being that you must be used to have to stay in the 3rd quarter But But For that Phyllis jacket wait So if you know what that is that Scarlet X 1 wall at 1 and is next to Whitefield what I know is that these 2 or less than 0 blissfully blue 0 and wired ones and wanted to Cynical 0 so happens if I add something less than 0 lessons their Reitzel unless a 0 you plus V X want to over why Plus Y Tu and protests here it lessons it and that's necessary the and therefore it's in the 3rd quarter because these court up to build negative 0 so that lies duplex feet is in the 3rd quarter So far so good looking like a vector space however The it fails of the scalar multiple The list to be the same rehab equals a negative to negative 3 The straw that there's negative to negative where now I have to build a multiply this by Amy scalar inhabits stay in the 3rd quarter so what and multiply by make it failed the negative number multiply by negative number of votes over the the 1st quarter that's not a vector space so this is not a vectors quadrant 3 heavily illustrated A crock of shit But But problem OK The fully was abstracted so far now have 3 things that we need would be responsible for only half the length of a vector Testing when near independence and now is this thing that I give you a vector space you'll see some the exact says in book related F who talk at next month's as a basis for saying that they could not language here a bassist basis for a vector space there so we know a vector spaces Just think of our to let your standard example in your mind the X-Y plane that standard that will be basis for the scanner remind you were of the Lord based on for what Think of it An English-language news or basis In this context what is the basis for that argument How's that were being used in that context What is the basis for your argument is not a phrase that we could use an English-language so what that means is was the foundation appear argument what's the logical components of argument graduates that the basis for what Why do you believe that that's a legitimate argument OK so in that context it means The foundation that's it's going to be electors ceases to be on firmer ride out the definition of all talk about this being a minimal set out purist Like you're little minimal set of logical arguments that supports Yuri supports your argument a minimal set of sectors that generate the Baltic states that How do It's a are you as the minimal set of vectors marital status go ahead now will move on Everybody got that now we have we an example focus sticker are standard example of a vector spaces are and what you wanna think of it is it's it's a bunch of sectors there at site later I can build wolves based from these 2 vectors here the 2 vectors and this is unique said could have other do this So let's go he won cuts canopied 1 0 and 2 these the Others even drop here years there's anyone Want and there's too so I mean by its a set of actors that generate boreholes space so it generates toll states that means that if a youth you give me any vector there I should be able to write it in terms of that's what it means and what we're lending by right in terms scandal linear combination is easy to see this as a big joke poll shows if how let's just take an example a stake of vector and rated in terms of
I have read week
His say to 3 . 0 rate to 3 in terms of well it's 2 times 1 0 allowed 3 times and will pursue this is 2 0 plus 0 3 and own a combined make it does is take any vector all states was supposed easygoing Tapie renowned all the little things that haven't specified what We are say they just by this argument that the same as they a kind 1 0 plus times 0 1 That's equally be so she you see how that that set of vectors making any space from those 2 that's what it means to be hike in there like Lego can billable space easily easiest wants to see the spread out over a 1st to of this as a linear combination that means to generate The horse from a basis to lay down nomination of the as you get everything in the space However would think of a basis when I give you the the 2 basis vector Is what it does is it creates new used to the city when you were a kid and they said grab this you said that you set their Irak season you put our little numbers on taxis sees you were setting up a basis you realize that you're setting up his grit so now if I want to 3 there's 2 and then 3 here So far I wanna get to that point here to 3 I need to be go over to hear NATO up there you're just moving along the basis as soon as I decided the basis is I get agreed for the holes space some grades are better than others all kind of explore that at some point but right now we're just getting the 16 8 6 of the bases But 22 days early to try to last time but I was very successful I might be successful today tried er The thing about our 2 is it that 2 dimension was the 1st time a connection between the dimensions of a vector space and then the space discounts So this is two-dimensional which means we need a minimum of 2 vectors to form a basis and that's Maybe you worst looking at this definition arrested a minimal set of actors I iconic glazed over there and I just talked about them being a set of vectors that generated the same thing about the animals so it in order to to generate a two-dimensional space I needed leased to vectors and those vectors work He won a need to because have other bases But Right now it is going to be OK let's let's go up What about are 3 OK let's just stick that frees extended to 3 dimensions are I don't have to Cars 3 years Three-dimensional OK so that if I needed to forge two-dimensional that I'm need 3 receive a minimum of 3 vectors for A basis for Cascilla tell what they should be here he won 1 0 0 sees can just do this bag extension he hears the 2 sectors that generate are in our 3 adept at 3 components and nearly 3 of them there's the 2nd 1 and the 3rd 1 If Off off a general was straight down what what we have that basis but this is linearly independent that He gave in all our and to set more than that and magic But sir
The ceiling example of how the So if something formed the basis then that means I can generate space for that sort minimal set of sectors that generate the whole space OK so what I can do is acting give you a vector 84 basis sectors and I can say right this sector terms of these 4 years an exam will take those vectors that we have a force space before we already had this example it's gone we are he showed that those early nearly independent fat 4 sectors that are linearly independent I should be able to write
We are very sorry they give in example 3 negative 2 0 1 As A linear combination of B want to be 3 D 4 What is that we have I do this follows taking some Mckeighan cancer said that can't be the answer Time focus aware that means a right as a linear combination or right now Orlando won 81 landed she needs to burst and this 3 D 3 landed 4 feet of water that was scary has a numerical value That guy is equal to that 3 negative G 2 0 Negative ones And then I wanna sulfur lamp so and no 1 is But 1 0 0 0 and June times 1 native water just 3 The 3 here 0 1 1 0 0 0 1 What In your goalless could find 1 The plan history Mn before all this is this is this is A system of linear equations with 4 area before variables Erland 1 landed 2 and 3 and 4 If you like you can put kaolinite matrix this is 3 negative 2 0 negative 1 and then you have 1 0 0 0 1 negative 1 0 0 0 1 1 EUR 0 0 Already see that's the augmented matrix to solve that system of equations anywhere I could tell you what theater he is equal to the tune of EUR on plus 22 minus carry last so that means that land that 1 or 2 to equals 1 380 negative wonderland was 1 that a good practice treated is seeking get that answer to system of equations We you can see that the last solved immediately doesn't give Yallamba 1 11 4 equals 1 you get that immediately from the bottom equation that you back substitute the Nederlander 3 they have landed 3 backs of certificate landed 2 years later the 2 of them once you knew that hero reduction just backs I toward erasing you can think about maybe asking questions to her students replaced
This is all a Section 10 . wanted actually leading out of stuff from 10 . 1 but you can't cut
You can count on a step by cover lectures the stuff to responsible for to some other stuff in that section while it might seem interesting your focusing on stuff for the exam stuff from lectures OK so where we had so far we've had The length of a vector Linear because the near independence Writing something as a linear common add a 3rd 1 in there and regard for things so far for the last concept of the day yes I did that to it If you solve this system here you're going to get laminar 1 equals 2 landed 2 and you put the Landers in any of that But Book 1st thing is a vocabulary lesson perpendicular own orthogonal are interchangeable terms perpendicular is Latin for a right angle an orthogonal is Greek for a rating 5 your orthodontist there that that's the person that makes you teeth irradiating goes to your job on the sea or doctors or those means rating rate increase data listen and you're probably used to this journey but the book fixed orthogonal that's what I want Says that just to make sure is up to date on the current vocabulary planning issue there were really studying perpendicularity it is a concept that about 2 vectors that's perpendicular Was Clairol that so the issue here is this is a specific case issue is it I have you and I have we here there's an angle between Morgan ado here's where there's a couple things going on we sort of interested in the angle not as a general concept religious 20 user-defined I want vectors are perpendicular but then the other thing that we haven't done which we did before with major cities as we did multiplication reggae Gaby 2 matrices you get Adam who could multiply the potentially We don't have any kind of concept of multiplication in this vector space so that's the 1st thing explore The inner product to that that's it can erase drive as a wise it is hereby requires adjust the product of 2 vectors while 2 vectors It's hard to do the product the 2 equals 1 2 And I have maybe because you won You have no idea the multiplication of those 2 If you want you to This is a tube I want that you want to That's also to buy 1 we can this So we can do is have regular products of 2 vectors McKinney said It was still not understand why the were introduced her 2nd person tell you what it is in order to combine 2 vectors in him Multiplication have to do it in a way that it works for matrix multiplication Imagine era I make this trans pose becomes 1 by to race them the choose match so that it would work for the inner product is going to be defined as you trams that's how I can get the multiplication work the 1st place than what happens if I do that well said that you want to here that's on its side now you transpose turns turns the Colvin row And manner that he is still in its bright column position but now I could do this matrix multiplication What is a tremendous honor aside here be won a new 1 match that I have you to in a way you don't really even have to turn it on its site when you're gonna do the inner fodder for attributed to match the top components Group a match of the 2nd component unit and then you have the result of what happened to you could write this out or you can just look at it and do it either way this is the definition of the product OK so like what I did before with matrix multiplication we talked about inside the Matrix oozes geometry
That explore what what is this enterprise was the intuition with the picture you should have in your mind when you do this
Algebraic manipulation of the er how It derived this but spare you that has history reproduce this picture years you need and what I want you to When I multiply these 2 vectors you might wanna think of it as multiplying the likes of the sectors that would be a legitimate thing if this satellite to the satellite of 380 might say what was going on Malta Pioline 2 times That's not quite what's going on it's a little modification of that reason it's called here Friday is ordered to Israeli take the shadow of you on that sort of the likes of you a long be remembered So this gathering is going to take a look at this little triangle rate But Indeed extends out further had this little shadow The shadow of you on TV and here's the length of of you and here's angle call shadow Khaled An extra set would assault with his ex Hey we all agree that cosine of data that's equal to the adjacent Over the hypotenuse we all agree on a prerequisite for the course E peso ahead so that it would see decent here it's X and the hypotenuse was the view of the matter lies the extra shadow said that's equal to the length of you Times across And what that computer for you
In the multiplication of the shadow of the sector and we could prove that is meant to take me a couple panels and that of the time I just won't take my word for it could also find in the book but here's here's the deal so that if I do
The length of feed At times the shadow That I have Notation for this sequel to the length of these and then the shadow is computed that scurvy That's the shadow it turns out that that is equal to this enterprise that's the way you wanna think of theater product is you're just multiplying the length but it's not quite the links it's the shadow of the length of 1 and the like Back Look at will Since this is a summary right you clear alphabetical order On talk and will now will exploit this because are all bull here's final when 2 vectors are perpendicular so that's a snow this year so
If outlets of the European So I suppose you and be will use their word orthogonal
Both drug picture Look the picture this year's and there's you and their perpendicular so that means the data equals 90 degrees For what's what's the cosine of 90 degrees off a prerequisite for this course closer 90 0 0 the unanswered here we see the X X Y coordinate is that it is 90 degrees X courted 0 there the cosigner cosigner Ohio close of 9 0 Castille Way This is 90 what does that mean that put that in that context over here so so if I have to win these guys are 0 rate because this is this is a vector you and that's a vector they got blank That supposedly equal to lead to transpose that's theater product but see when I have a 90 degrees than this is 0 so that had implies you tranche was was And that gives us a test so now I give you 2 vectors and SAR orthogonal you go in you take the inner product and if you find out that 0 then yes they are perpendicular and if they're not 0 than not they're not this test tolerate that sector Codify that I will do couple examples of only 1 6 Blackshaw
OK so you and we end any dimension a dimension
1st of all let's say would But say you and you it is I don't know what you want to review and Angelia is not to be not by do theater Prada You're Trans posed what is that lets you want you and you want to end and then you just do this matrix multiplication that essentially all you doing is just matching the 1st components the 2nd down the last 1 more so this is eagerly you want 81 Plus you Giambi to plus That's the inner product fur and mention but the perpendicularity still the same thing because it's still just 2 vectors so there's still knowledge in 83 freezer for space is still at an angle between them and when they're perpendicular For them the shadow 0 rate perpendicular is the shadow 0 So that'll still be true in our here you When you get 0 for the inner product that implies do the same if and only if you indeed are orthogonal OK so what's a typical questions that you get On an exam with regards to to this 1 like this 1 here which sectors But have some you're not wound is sustained victors boat move clear So there's 3 vectors say which ones orthogonally can graph on you can't see him so you have to use this test yet to theater product if you get 0 there a orthogonal you don't 0 they're not our thoughts and test so need 1 transpose too That skin and be 1 times already all out but it's issue plus 0 plus 0 plus 4 leader not our thought so you might be say noted duty 1 trainers Posey 2 1 you transpose you want this thing if switch the worry are distilled into 2 times 1 0 times on negative or understood as a matter the just picked too Paramount by the inner product that it 0 they're orthogonal not to help that's also could to trade was 1 of the same thing Payless check 1 transpose 3 The madam a negative 1 plus 0 minus 1 plus That is And that's also equal to these 3 China's close to that is 0 so be 1 is perpendicular to these 3 orthogonal than the last 1 I had pictures 3 so which 1 happen again I haven't anti-Jewish 83 that's negative to 0 Plus 0 plus 1 Negative ones which is not equal to 0 will be futile and are not a lot of alive time and a steady if you pay attention notes of this class because today's lecture particularly because there's a think 5 things that are very specific but you have to to know that you a Henry chided this I regard this is 1 of writing things in terms of a basis is another 1 that vector space is a vector space or not that's that's a 3rd 1 than there was like the vector And then there was a linear independence to determine the nature Texas 5 thinks so I talked a lot basically is as these 5 things you have to to be responsible for the lecture and if you know how to do these examples that I didn't class you're ready that's another thing will save you time you just keep practicing these examples of reading for wherever I put on the exam from Section 10 . 1 assault 1 more Consolata do today you get them today but I have 1 more concept from Section 10 . 1 0 cover that on Monday that will detection 10 . 2 big these big section 1 is 10 . 2 is another big Section 10 . 3 is not big will be ready for the big thing now I will tell you because of legal nervous about the exam on Wednesday I'll give you a sample of him throughout the it'll be based I would say it's a sample midterm it's it's like sample midterm questions that much longer than that you'll have all the things that I think errors that he should be responsible for any examples of each 1 of those things for you practice for a couple of days before you get the actual test on Friday any questions for you guys Bulls I my post among others Chalkboard a bit better than just like the real midterm you don't get into the jitters that sometimes work But I'll do some woman classroom Wednesday and my that Duval test that on not just see what we're kind Urgent But this On Monday a minute just 2 1 last concept from 10 1 ordered tend to and then 10 on 10 3 of the it's just a quick What Yet it is question just That this mid-term finally Gap Most of the sketches that I do in this class Our for your intuition It's not the algebras work It is necessary but it's weird to me a presenting algebra without having these pictures there there anyway but I'll tell you that You know OK I believe
Matrizenrechnung
Subtraktion
Einfügungsdämpfung
Punkt
Natürliche Zahl
Wasserdampftafel
Hausdorff-Dimension
Klasse <Mathematik>
Gruppenoperation
Zahlenbereich
Gleichungssystem
Term
Raum-Zeit
Eins
Variable
Reelle Zahl
Perspektive
Lineare Geometrie
Gleitendes Mittel
Funktion <Mathematik>
Dimension 2
Erweiterung
Matrizenring
Matching <Graphentheorie>
Mathematik
Inverse
Mathematisierung
Kombinator
Lineare Gleichung
Physikalisches System
Vektorraum
Rechter Winkel
Garbentheorie
Ordnung <Mathematik>
Aggregatzustand
Subtraktion
Sortierte Logik
Hausdorff-Dimension
Vektorraum
Raum-Zeit
Länge
Verschlingung
Zahlenbereich
Vektorraum
Ausdruck <Logik>
Eins
Zahlensystem
Pythagoreisches Zahlentripel
Quadratzahl
Betrag <Mathematik>
ARCH-Prozess
Reelle Zahl
Rechter Winkel
Pythagoreischer Lehrsatz
Delisches Problem
Kantenfärbung
Abstand
Wurzel <Mathematik>
Gerade
Länge
Punkt
Sterbeziffer
Hausdorff-Dimension
Gruppenkeim
Vektorraum
Ereignishorizont
Dreieck
Physikalische Theorie
Knotenmenge
Quadratzahl
Einheit <Mathematik>
Rechter Winkel
Basisvektor
Pythagoreischer Lehrsatz
Zusammenhängender Graph
Wurzel <Mathematik>
Abstand
Koordinaten
Dimension 2
Länge
Grothendieck-Topologie
Sterbeziffer
Hausdorff-Dimension
Wellenlehre
Vektorraum
Vektor
Raum-Zeit
Ausdruck <Logik>
Vektoranalysis
Negative Zahl
Quadratzahl
Körper <Physik>
Zusammenhängender Graph
Wurzel <Mathematik>
Graphentheorie
Analogieschluss
Matrizenrechnung
Länge
Multiplikation
Quader
Hausdorff-Dimension
Gruppenkeim
Zahlenbereich
Parallelogramm
Skalarfeld
Raum-Zeit
Ausdruck <Logik>
Multiplikation
Negative Zahl
Regulärer Graph
Reelle Zahl
Zeitrichtung
Zusammenhängender Graph
Addition
Matrizenring
Mathematik
Kategorie <Mathematik>
Dimensionsanalyse
Vektorraum
Rechnen
Mereologie
Diagonale <Geometrie>
Zentrische Streckung
Exakter Test
Kombinator
Garbentheorie
Vektorraum
Teilbarkeit
Linearisierung
Verschiebungsoperator
Länge
Multiplikation
Zahlenbereich
Parallelogramm
Kombinator
Vektorraum
Diagonale <Geometrie>
Skalarfeld
Linearisierung
Ebene
Wellenpaket
Menge
Exakter Test
Hausdorff-Dimension
Stochastische Abhängigkeit
Gerichtete Größe
Vektorraum
Gerade
Richtung
Ebene
Subtraktion
Stochastische Abhängigkeit
Hausdorff-Dimension
Vektorraum
Analogieschluss
Richtung
Ebene
Sortierte Logik
Subtraktion
Konditionszahl
Hausdorff-Dimension
Bilinearform
Gerade
Hausdorff-Dimension
Konditionszahl
Vektorraum
Ordnung <Mathematik>
Tropfen
Konstante
Sterbeziffer
Exakter Test
Reelle Zahl
Güte der Anpassung
Zusammenhängender Graph
Kombinator
Vektorraum
Ordnung <Mathematik>
Eins
Linearisierung
Matrizenrechnung
Matrizenring
Punkt
Streuung
Zahlenbereich
Kombinator
Gleichungssystem
Parallelogramm
Physikalisches System
Vektorraum
Raum-Zeit
Multiplikation
Negative Zahl
Differenzkern
Minimum
Zusammenhängender Graph
Diagonale <Geometrie>
Gerade
Resultante
Matrizenrechnung
Subtraktion
Hausdorff-Dimension
Natürliche Zahl
Klasse <Mathematik>
Abgeschlossene Menge
Gleichungssystem
Zahlensystem
Exakter Test
Radikal <Mathematik>
Zustand
Zusammenhängender Graph
Spezifisches Volumen
Gerade
Analysis
Umwandlungsenthalpie
Beobachtungsstudie
Einfach zusammenhängender Raum
Lineare Abhängigkeit
Mathematik
Determinante
Matching <Graphentheorie>
Physikalischer Effekt
Stochastische Abhängigkeit
Kombinator
Vektorraum
Physikalisches System
Flüssiger Zustand
Dreiecksmatrix
Gerade
Linearisierung
Flächeninhalt
Rechter Winkel
Lineare Abhängigkeit
Länge
Subtraktion
Exakter Test
Menge
Matching <Graphentheorie>
Mathematik
Einheit <Mathematik>
Klasse <Mathematik>
Vektorraum
Innerer Punkt
Zahlenbereich
Kartesische Koordinaten
Element <Mathematik>
Kardinalzahl
Raum-Zeit
Skalarfeld
Erwartungswert
Multiplikation
Reelle Zahl
Meter
Radikal <Mathematik>
Optimierung
Drei
Einfach zusammenhängender Raum
Determinante
Spieltheorie
Ähnlichkeitsgeometrie
Vektorraum
Dreiecksmatrix
Dreieck
Teilbarkeit
Summengleichung
Sortierte Logik
Konditionszahl
Mereologie
Diagonale <Geometrie>
Aggregatzustand
Länge
Abstimmung <Frequenz>
Kartesische Koordinaten
Mathematische Logik
Term
Raum-Zeit
Skalarfeld
Eins
Negative Zahl
Multiplikation
Exakter Test
Zusammenhängender Graph
Drei
Schnitt <Graphentheorie>
Parametersystem
Grothendieck-Topologie
Extremwert
Stochastische Abhängigkeit
Kombinator
Vektorraum
Fokalpunkt
Linearisierung
Menge
Rechter Winkel
Basisvektor
Aggregatzustand
Standardabweichung
Punkt
Sterbeziffer
Extrempunkt
Hausdorff-Dimension
Kondition <Mathematik>
Zahlenbereich
Basis <Mathematik>
Extrempunkt
Term
Raum-Zeit
Gradient
Stützpunkt <Mathematik>
Zusammenhängender Graph
Dimension 2
Nominalskaliertes Merkmal
Einfach zusammenhängender Raum
Parametersystem
Erweiterung
Extremwert
Kombinator
Vektorraum
Forcing
Menge
Sortierte Logik
Basisvektor
Ordnung <Mathematik>
Aggregatzustand
Matrizenrechnung
Wasserdampftafel
t-Test
Zahlenbereich
Kombinator
Gleichungssystem
Physikalisches System
Fokalpunkt
Ordnungsreduktion
Eins
Lineares Gleichungssystem
Linearisierung
Variable
Flächeninhalt
Rechter Winkel
Garbentheorie
Schnitt <Graphentheorie>
Resultante
Matrizenrechnung
Länge
Prozess <Physik>
Ortsoperator
Sterbeziffer
Gruppenkeim
Term
Überlagerung <Mathematik>
Multiplikation
Einheit <Mathematik>
Zusammenhängender Graph
Umwandlungsenthalpie
Grothendieck-Topologie
Matrizenring
Matching <Graphentheorie>
Winkel
Stochastische Abhängigkeit
Orthogonale Funktionen
Strömungsrichtung
Vektorraum
Physikalisches System
Biprodukt
Skalarproduktraum
Linearisierung
Arithmetisches Mittel
Rechter Winkel
Garbentheorie
Lateinisches Quadrat
Ordnung <Mathematik>
Ext-Funktor
Geometrie
Multiplikation
Länge
Sterbeziffer
Menge
Sortierte Logik
Winkel
Abschattung
Vektorraum
Trigonometrische Funktion
Zahlensystem
Länge
Verschlingung
Rechter Winkel
Orthogonale Funktionen
Abschattung
Fortsetzung <Mathematik>
Vektorraum
Ordnung <Mathematik>
Biprodukt
Arithmetisches Mittel
Erweiterung
Sterbeziffer
Exakter Test
Hausdorff-Dimension
Abgeschlossene Menge
Orthogonale Funktionen
Vektorraum
Biprodukt
Trigonometrische Funktion
Matrizenrechnung
Lineare Abhängigkeit
Graph
Sterbeziffer
Winkel
Natürliche Zahl
Klasse <Mathematik>
Orthogonale Funktionen
Vektorraum
Biprodukt
Skalarproduktraum
Term
Raum-Zeit
Stichprobenfehler
Überlagerung <Mathematik>
Eins
Multiplikation
Negative Zahl
Exakter Test
Basisvektor
Stichprobenumfang
Abschattung
Zusammenhängender Graph
Garbentheorie

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Titel Math for Economists - Lecture 6
Serientitel Math for Economists
Teil 6
Anzahl der Teile 15
Autor Kronewetter, Jason
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nicht-kommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben.
DOI 10.5446/12900
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2013
Sprache Englisch

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Fachgebiet Mathematik

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