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And essentially when I'm talking about the linear combination of only one other vector, it's just multiplying it by a scalar. Well, let's think about that. There's multiple ways you can show this, but the easiest way is, well, look. To go from this entry to that entry, I'm just multiplying by 1. But if I multiply this ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
To go from this entry to that entry, I'm just multiplying by 1. But if I multiply this whole vector times 1, then I'm going to get a 2 here and I'm going to get a 3 here. So it won't work. Let me put it this way. If I want to represent this guy as a scalar multiple of that guy, so any scalar multiple of 1, 2, 3 is goin... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me put it this way. If I want to represent this guy as a scalar multiple of that guy, so any scalar multiple of 1, 2, 3 is going to be equal to 1c, 2c, 3c. And so we're saying this guy has to be represented somehow like that. If we say that this guy is somehow a scalar, it somehow can be represented by that guy. So... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If we say that this guy is somehow a scalar, it somehow can be represented by that guy. So that would have to be equal to 1, 1, 4. When you look at this top entry, it implies that c would have to be equal to 1. When you just, c is equal to 1. But when you look at this second entry, you think that c would have to be equ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
When you just, c is equal to 1. But when you look at this second entry, you think that c would have to be equal to 1 half. So you get a contradiction over here. c would have to be equal to 4 thirds. So there's no c where this will work. There's no multiple of c, and you can work that both ways. So there's no way that y... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
c would have to be equal to 4 thirds. So there's no c where this will work. There's no multiple of c, and you can work that both ways. So there's no way that you can represent one of these guys as a linear combination of the other. And you can actually prove other ways, maybe more formally, that this is linearly indepe... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So there's no way that you can represent one of these guys as a linear combination of the other. And you can actually prove other ways, maybe more formally, that this is linearly independent. But given that this is linearly independent, I think you're satisfied with that. We can then say that the set of vectors 1, 2, 3... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
We can then say that the set of vectors 1, 2, 3, and 1, 1, 4, this is a basis for the column span of A. Now, I'm going to let you go in this video because I think I've gone well over time. But what I'm going to do in the next few videos is now that I've established that this is a basis for the column span of A, we can ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And we can think about what the span of those two vectors are. We're going to see that it's a plane in R3. Span of 1, 1, 4. And just as a quick reminder, I said it a couple of times. I said it's a basis. All I'm saying is that these guys, they both span the column space of A. When I had four vectors, they also span the... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And just as a quick reminder, I said it a couple of times. I said it's a basis. All I'm saying is that these guys, they both span the column space of A. When I had four vectors, they also span the column space of A. But what makes them a basis is that these guys are linearly independent. There's no extra information or... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
We used these exact examples where if this was matrix A, I just took it and I put it in reduced row echelon form, and I figured out which of these columns in my reduced row echelon form of A are pivot columns. It turned out to be the first one, the second one, and the fourth one. Then the method is, you say, look, the ... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
Since they form the basis, and if you want to know the dimension of your basis of your column space, which is also called the rank, you just say, well, there's three in there. It has a rank of 1, 2, 3. In this video, I want to discuss a little bit about why this worked. Why were we able to just take the corresponding c... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
Why were we able to just take the corresponding columns? Why did linear independence of these three guys imply linear independence of these three guys? Why was the fact that I can represent these guys, this guy right here is a linear combination of these three, or this guy is a linear combination of these three, why do... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
The first thing that wasn't too much of a stretch of the imagination in the last video was the idea that these pivot vectors are linearly independent, so R1, R2, and R4. Everything I'm doing, I'm kind of applying to this special case just so that it's easier to understand, but it should be generalizable. In fact, it de... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
That's because the very nature of reduced row echelon form is that you are the only pivot column that has a 1 in that respective row. The only way to construct it is with that vector. You can't construct it with the other pivot columns because they're all going to have 0 in that row. When I say it's linearly independen... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
When I say it's linearly independent, I'm just saying the set of pivot columns. Let me say this in general. The set of pivot columns for any reduced row echelon form matrix is linearly independent. It's just a very straightforward argument because each column is going to have 1 in a very unique place. All of the other ... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
It's just a very straightforward argument because each column is going to have 1 in a very unique place. All of the other pivot columns are going to have a 0 in that same place, so you can't take any linear combination to get to that 1 because 0 times anything minus or plus 0 times anything can never be equal to 1. I t... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
That means that the solution to C1 times R1 plus C2 times R2 plus C4 times R4, the solution to this equation, because these guys are linearly independent, we know that this only has one solution and that's C1, C2, and C4 is equal to 0. That's the only solution to that. Another way we could say it is if we write R times... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
This will be some special member of your null space. It's a particular solution to the equation. This is equal to 1, 2, 3, 4, 0 because we have 4 rows here. Now, if we just expand this out, if we just multiply 1 times C1 plus 0 times C2 minus 1 times 0 plus 4 times 0, you'll get... Actually, let me think. A better way ... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
Now, if we just expand this out, if we just multiply 1 times C1 plus 0 times C2 minus 1 times 0 plus 4 times 0, you'll get... Actually, let me think. A better way to explain it. This multiplication right here can be written as, and we've seen this multiple times, C1 times R1 plus C2 times R2 plus 0 times R3, so we coul... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
That's R5 right there. All of that equal to 0. The only solution to this, because we know that these 3 columns are linearly independent, or the set of just those 3 columns, those 3 pivot columns are linearly independent, the only solution here is all of these equal to 0. That's exactly what I said right up here. The on... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
That's exactly what I said right up here. The only solution here, where if these 2 are 0, is that these guys also all have to equal 0, if I already constrain these 2. Now, the one thing that we've done over and over again, we know that the solution set of this equation, the solution set of Rx is equal to 0, is the same... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
How do we know that, or what do I mean? The solution set of this is just the null space. The solution set is just the null space of R. It's all of the x that satisfy this equation, and we know that that is equal to the null space of A, where R is just A in reduced row echelon form. The null space of A is all of the x's... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
The null space of A is all of the x's that satisfy this equation. Now, the only version of this that satisfied this equation was when C1, C2, and C4 are equal to 0. That tells us that the only version of this, C1, C2, 0, C4, 0, that satisfies this equation, or this equation, is when C1, C2, and C4 is equal to 0. Anothe... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
Another way of saying that, if this is vector A1, A2, A4 right here, if you multiply this out, you get C1 times A1 plus C2 times A2, and then 0 times A3 plus C4 times A4 is equal to 0. Now, these guys are going to be linearly independent if and only if the only solution to this equation is they all equal to 0. Well, we... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
The only solution to this was, if I go ahead and I constrain these two terms to being equal to 0, the only solution to this is all of these C's have to be 0. Likewise, if I constrain these to be 0, the only solution to this is that C1, C2, and C4 have to be 0. Those guys have to be 0, which imply that these three vecto... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
We're halfway there. We've shown that, look, because the pivot columns here are linearly independent, we can show that they have the same solution set. The null space of the reduced row echelon form is the same as the null space of our original matrix. We were able to show that the only solution to C1 times this plus C... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
We were able to show that the only solution to C1 times this plus C2 times this plus C4 times this is when all the constants are 0, which shows that these three vectors, or a set of those three vectors, are definitely linearly independent. The next thing to prove that they are a basis is to show that, look, all of the ... | Showing relation between basis cols and pivot cols Linear Algebra Khan Academy.mp3 |
And the function just maps each of the specific entries of x to an entry in y. When I say map, it really just creates an association. If we think of these in maybe even less abstract terms, on some levels it's more abstract, you could view x as a basket of bananas and y as a basket of apples. And for every banana, you'... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
And for every banana, you're associating it with one of the apples. That would be the definition, the mapping of going from each of those bananas to each of those apples would be a function. I don't know if that helps you or not. But I just want to kind of broaden your already preconceived notion of what a function is.... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
But I just want to kind of broaden your already preconceived notion of what a function is. I mean, everything that you've probably seen before probably took a form that looks something like that, where you said, oh, a function is just give me some number and I'll give you another number, or I'll do something to that nu... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It's association between any member of one set and some other members of another set. Now, we know that vectors are members of sets. Vectors. In particular, if we say that some vector x is a member of some set, let me just say it's a member of Rn, because that's what we deal with, all that means is that this is just a ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
In particular, if we say that some vector x is a member of some set, let me just say it's a member of Rn, because that's what we deal with, all that means is that this is just a particular representation of an n-tuple. Remember what Rn was. Rn we defined way back, I think, maybe at the beginning of the linear algebra p... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
We defined it as the set of all n-tuples, x1, x2, xn, where your x1s, x2s, all the way to xn's are a member of the real numbers. So your Rn is most definitely a set. This could be Rn, and obviously the use of the letter n is arbitrary. It could be Rm, it could be Rs. Rn is just kind of a placeholder for how many tuples... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It could be Rm, it could be Rs. Rn is just kind of a placeholder for how many tuples we have. This could be R5, it could be 5 tuples. When we say that a vector x is a member of Rn, we're just saying that it's another way of writing one of these n-tuples. All of our vectors so far were our column vectors. That's the onl... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
When we say that a vector x is a member of Rn, we're just saying that it's another way of writing one of these n-tuples. All of our vectors so far were our column vectors. That's the only type that we've defined so far. We say it's this ordered list where each of the members are a member of R. It's an ordered list of n... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
We say it's this ordered list where each of the members are a member of R. It's an ordered list of n components, x1, x2, all the way to xn, where each of those guys, where each of those x1s, x2s, all the way to xn's, are a member of the real numbers. That's by definition what we mean when we say that x is a member of R... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's say that this set right here is Rn. Then let me just change, just to be general, let me create another set right there and call that set right there Rm. Just a different number. It could be the same as n, it could be different. This is m-tuples, that's n-tuples. The vectors we've defined, that vectors can be memb... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It could be the same as n, it could be different. This is m-tuples, that's n-tuples. The vectors we've defined, that vectors can be members of Rn. So you could have some vector here, and then if you associate with that vector in Rm, if you associate it with some vector in Rm, if you associate it with, let's call that v... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
So you could have some vector here, and then if you associate with that vector in Rm, if you associate it with some vector in Rm, if you associate it with, let's call that vector y, if you make this association, that too is a function. That might have already been obvious to you. This would be a function that's mapping... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Actually, I just want to make one little special note here. When I just drew the arrow like this, this shows that I'm mapping between two sets. I'm taking elements of this set and I'm associating with them elements of that set. In the last video, you probably saw this. I want to do the side note because I realize it mi... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
In the last video, you probably saw this. I want to do the side note because I realize it might have been confusing. I introduced you to another way of writing a function like this, where I said f could be defined as a mapping for any given x to x squared. I just want to make a note on the notation. When I just have a ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
I just want to make a note on the notation. When I just have a regular arrow, I'm going between sets. When I have this little vertical line at the base of the arrow, that's kind of the function definition. It tells me for any x you give me in the first set, in the second set I'm going to associate this x with, in this ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It tells me for any x you give me in the first set, in the second set I'm going to associate this x with, in this case, x squared. I just wanted to make that side note. The whole direction I was going in is that vectors are valid elements of sets. Functions are just mappings between elements of sets. You could have fun... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Functions are just mappings between elements of sets. You could have functions of vectors. I even touched on that a little bit in the last video when I talked about vector-valued functions. If your codomain is a subset of our m, where m is greater than 1, then we say your function is vector-valued. It's not just mappin... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
If your codomain is a subset of our m, where m is greater than 1, then we say your function is vector-valued. It's not just mapping into the real numbers. It's mapping into some m-tuple of real numbers. If you mapped to two-dimensional space, you're dealing with a vector-valued function. I've been all abstract and what... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
If you mapped to two-dimensional space, you're dealing with a vector-valued function. I've been all abstract and whatnot, so let me actually deal with some vectors, and it might make everything a little bit more concrete. Let's say I define the function f as f of x1, x2, and x3 is equal to x1 plus 2x2, and the second c... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
I actually haven't formally defined coordinates for you yet, but I think you understand that just from your basic algebra training. Let's say that that's my function definition. Based on the notation that we've been introduced to, we could say that f is a mapping, its domain is R3, and it maps from R3, or it associates... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
This is a 2-tuple. This is in R2. This is a 3-tuple. Or another way we could do this, if we just wanted to write it in vector notation, I could write that if you pass f to f, some vector, x1, x2, x3, I could say this will be equal to the vector, and now it's going to have a two-component vector. It's going to be a vect... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Or another way we could do this, if we just wanted to write it in vector notation, I could write that if you pass f to f, some vector, x1, x2, x3, I could say this will be equal to the vector, and now it's going to have a two-component vector. It's going to be a vector in R2, where the first term is x1 plus 2x2, and th... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
See what it does to the vectors. What is f of the vector 1, 1, 1? Well, I get 1 plus 2 times 1 is I get the vector 3, and then my second term is just 3 times this one, so I get the vector 3, 3. Fair enough. Let me do another one. Just to really experiment with this mapping, if I take the f of the vector in R3, 2, 4, 1,... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Fair enough. Let me do another one. Just to really experiment with this mapping, if I take the f of the vector in R3, 2, 4, 1, what do I get? That equals 2 plus 2 times 4. That goes to the vector 10. I get 2 plus 2 times 4, and then 3 times the third term right there, so the vector 10, 3. How can we visualize this? | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
That equals 2 plus 2 times 4. That goes to the vector 10. I get 2 plus 2 times 4, and then 3 times the third term right there, so the vector 10, 3. How can we visualize this? Well, three-dimensional vectors or vectors in R3 are not always the easiest to visualize, but I think we can attempt to visualize these two guys.... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
How can we visualize this? Well, three-dimensional vectors or vectors in R3 are not always the easiest to visualize, but I think we can attempt to visualize these two guys. Let's see. The first, let me do it a little bit better than that. Let's say that this is the x1 axis, that's the x2 axis, that's the x3 axis. This ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
The first, let me do it a little bit better than that. Let's say that this is the x1 axis, that's the x2 axis, that's the x3 axis. This first vector right here, this yellow one, 1, 1, 1, it'll look like this, 1, 1, 1. If I were to go out here, then go out here, then go up 1, the point would be right there, and if I wer... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
If I were to go out here, then go out here, then go up 1, the point would be right there, and if I were to draw it in standard position, I'd start at the origin, and the vector looks something like that. Then the second guy, 2, 4, 1, it would look like this. It would be, we'd go 2 out here, we'd go 4 in this direction,... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
2, 4, 1. I think you get the idea. I've drawn these two vectors, these two vectors that are essentially in my domain. Our domain is R3. This is R3 right here. Let's see what our function maps these vectors to. It maps, what's our codomain? | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Our domain is R3. This is R3 right here. Let's see what our function maps these vectors to. It maps, what's our codomain? Our codomain is R2, so this is much easier to visualize for us. We just have to draw two axes, just have to draw two axes like this. Let's call this X1, and let's call this X2. | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It maps, what's our codomain? Our codomain is R2, so this is much easier to visualize for us. We just have to draw two axes, just have to draw two axes like this. Let's call this X1, and let's call this X2. What does F of 1, 1, 1 of this yellow vector, it becomes 3, 3. If I do it in yellow, 1, 2, 3, 1, 2, 3. It gets, y... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's call this X1, and let's call this X2. What does F of 1, 1, 1 of this yellow vector, it becomes 3, 3. If I do it in yellow, 1, 2, 3, 1, 2, 3. It gets, you mean this one. If I draw it in standard position, the vector looks like this. We literally, our function went from, mapped from this vector in R3 to this vector... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It gets, you mean this one. If I draw it in standard position, the vector looks like this. We literally, our function went from, mapped from this vector in R3 to this vector in R2. That was what our function did. Likewise, if we take the other vector, we went from this 2, 4, 1 vector to this vector 10, 3. 1, 2, 3, 4, 5... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
That was what our function did. Likewise, if we take the other vector, we went from this 2, 4, 1 vector to this vector 10, 3. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. It's going to look something like this. It's going to be 3 up, so it's going to look something like this. This vector right here by our function F got mapped, let ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It's going to look something like this. It's going to be 3 up, so it's going to look something like this. This vector right here by our function F got mapped, let me do a different color, got mapped to this vector. This vector right here in R3 got mapped to this vector in R2 by our function. This is just a switch of te... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
This vector right here in R3 got mapped to this vector in R2 by our function. This is just a switch of terminology. When we talk about functions of vectors, the term that we tend to use is the word transformation. It really is the exact same thing as a function. I don't want to confuse you because if you've watched the... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
It really is the exact same thing as a function. I don't want to confuse you because if you've watched the differential equations playlist, you saw the idea of a Laplace transformation, which is really an operation that takes a function as an argument. In this case, and when we're dealing in the linear algebra world, a... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
The general notation, instead of writing a lowercase f like that, for a function, people use an uppercase T to say it's a transformation. It doesn't have to be an uppercase T, but that's the one that people use the most. This could be a G or an H, but people always use a lowercase f. The same way we could have written ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
My sense of why in the linear algebra world they use this is because you kind of imagine that this vector is being changed into that vector or that this vector is being transformed into that vector. I think that's why they call it a transformation as opposed to a function. It actually makes a lot more sense when you st... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
We'll talk a lot more about that in the future. I just want to introduce you to this notation. These statements, I could have just as easily written my, I could have replaced all my f's with t's and I could have defined some transformation. I just want to make you comfortable with the notation. I could have defined it ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
I just want to make you comfortable with the notation. I could have defined it similarly a transformation from R3 to R2. I could have said that t of x1, x2, x3 is equal to the 2-tuple x1 plus 2x1, 3x3. I could have just as similarly put a t up here because I've defined it the same way. I could have said t of my vector ... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
I could have just as similarly put a t up here because I've defined it the same way. I could have said t of my vector 1, 1, 1 is equal to 3, 3. Now you might say, hey Sal, why are you going through all this trouble of replacing the t's with s? I'm just doing this so you don't get confused. So that when you see, in your... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm just doing this so you don't get confused. So that when you see, in your linear algebra book, when you see linear algebra problems where you see this big capital T and you're like, wow, I've never seen that before and they're using this fancy word called a transformation, this is completely identical to your notion... | Vector transformations Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's think of an example of what wouldn't and what would be a vector. So if someone tells you that something is moving at five miles per hour, this information by itself is not a vector quantity. It's only specifying a magnitude. We don't know what direction this thing is moving five miles per hour in. So this righ... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
We don't know what direction this thing is moving five miles per hour in. So this right over here, which is often referred to as a speed, this is a speed, is not a vector quantity just by itself. This is considered to be a scalar quantity. If we want it to be a vector, we would also have to specify the direction. So fo... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
If we want it to be a vector, we would also have to specify the direction. So for example, someone might say it's moving five miles per hour east. So let's say it's moving five miles per hour due east. So now this combined five miles per hour due east, this is a vector quantity, and now we wouldn't call it speed anymor... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
So now this combined five miles per hour due east, this is a vector quantity, and now we wouldn't call it speed anymore, we would call it velocity. So velocity is a vector. We're specifying the magnitude, five miles per hour, and the direction east. But how can we actually visualize this? So let's say we're operating i... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
But how can we actually visualize this? So let's say we're operating in two dimensions, and what's neat about linear algebra is obviously a lot of what applies in two dimensions will extend to three, and then even four, five, six, as many dimensions as we want. Our brains have trouble visualizing beyond three, but what... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
But let's just go back to our straight traditional two-dimensional vector right over here. So one way we could represent it, as an arrow that is five units long, we'll assume that each of our units, our unit here is miles per hour, and that's pointed to the right, where we'll say the right is east. So for example, I co... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
The length of the arrow specifies the magnitude, so one, two, three, four, five, and then the direction that the arrow is pointed in specifies its direction. So this right over here could represent visually this vector. If we say that the horizontal axis is say east, or the positive horizontal direction is moving in th... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now what's interesting about vectors is that we only care about the magnitude and the direction. We don't necessarily care where we start, where we place it when we think about it visually like this. So for example, this would be the exact same vector, or it would be equivalent vector to this. This vector has the same ... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
This vector has the same length, so it has the same magnitude, has a length of five, and its direction is also due east, so these two vectors are equivalent. Now one thing that you might say is, well, that's fair enough, but how do we represent it with a little bit more mathematical notation so we don't have to draw it... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
If you're publishing a book, you can bold it, but when you're doing it in your notebook, you would typically put a little arrow on top of it, and there's several ways that you could do it. You could literally say, hey, five miles per hour east, but that doesn't feel like you can really operate on that easily. The typic... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
So for example, this one only moves in the horizontal dimension, and so we'll put our horizontal dimension first, so you might call this vector five comma zero. It's moving five, positive five, in the horizontal direction, and it's not moving at all in the vertical direction. And the notation might change. You might al... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
You might also see notation, and actually in a linear algebra context, it's more typical to write it as a column vector like this, five, zero. This, once again, the first coordinate represents how much we're moving in the horizontal direction, and the second coordinate represents how much are we moving in the vertical ... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
You could have other vectors. You could have a vector that looks like this. It's moving three in the horizontal direction, and positive four, so one, two, three, four, in the vertical direction. So let me see. It might look something like this. So this could be another vector right over here. Maybe we call this vector ... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me see. It might look something like this. So this could be another vector right over here. Maybe we call this vector vector A, and once again, I want to specify that it is a vector, and you see here that if you were to break it down, in the horizontal direction, it is moving three, or it's shifting three in the... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
Maybe we call this vector vector A, and once again, I want to specify that it is a vector, and you see here that if you were to break it down, in the horizontal direction, it is moving three, or it's shifting three in the horizontal direction, and it's shifting four, positive four, in the vertical direction, and we get... | Vector intro for linear algebra Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's a very useful tool. And that just told us if I have two vectors, x and y, they're both members of Rn, and they're both non-zero vectors. And that was an assumption we had to make when we did the proof. Otherwise, there's a potential of dividing by one of their magnitudes, so that would have been a big no-no. But i... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Otherwise, there's a potential of dividing by one of their magnitudes, so that would have been a big no-no. But if we assume that they're both non-zero, then we can say that the absolute value of their dot products is going to be less than or equal to the products of their individual lengths. So that's the length of ve... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then this is the length of vector y. And of course, this is just a regular number, and then each of these are just regular numbers. They're not vectors once you take a length. The length of a 50-dimensional vector could just be the number 3. It's just a scalar value. So this is just scalar multiplication here. And ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
The length of a 50-dimensional vector could just be the number 3. It's just a scalar value. So this is just scalar multiplication here. And we also learned that the only time that this inequality turns into an equality is a situation where x is equal to some scalar multiple of y. And so in some textbooks, you'll say, a... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And we also learned that the only time that this inequality turns into an equality is a situation where x is equal to some scalar multiple of y. And so in some textbooks, you'll say, and this has to be a non-zero scalar multiple, but that's a bit obvious. I told you that x and y are non-zero. So if this was zero, then ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So if this was zero, then x would be zero. And I just said that x is non-zero, but if you want to say that, you could say that c also is going to be non-zero. But that essentially just falls out of this information there. But if this is the case, and if and only if this is the case, then we can say that the absolute va... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
But if this is the case, and if and only if this is the case, then we can say that the absolute value of the dot product of the two vectors is equal to the product of their lengths. Now, this is all just a review of what I did in the last video. Now, what else can we do that's useful with it? So let's just play around ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's just play around a little bit. But I can't claim to be experimenting. I know where this is going to go. Let's see what happens if I were to take the length of x plus y. So I'm going to add the two vectors and then take the length of that vector squared. Well, we know from a couple of videos ago that the length... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's see what happens if I were to take the length of x plus y. So I'm going to add the two vectors and then take the length of that vector squared. Well, we know from a couple of videos ago that the length squared can also be rewritten as the dot product of a vector with itself. This right here, x plus y, I know it l... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
This right here, x plus y, I know it looks like two vectors, but it's two vectors added to each other. So it's really a vector. x plus y is a real vector. I could graph x plus y. So the length of x plus y squared, I can rewrite it as the dot product of that vector with itself. So x plus y dot x plus y. And all of these... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could graph x plus y. So the length of x plus y squared, I can rewrite it as the dot product of that vector with itself. So x plus y dot x plus y. And all of these are vectors. These aren't just numbers. And this is the dot product. It's just not normal multiplication. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And all of these are vectors. These aren't just numbers. And this is the dot product. It's just not normal multiplication. But we saw two videos ago that the dot product has a distributive and the associative and the commutative properties, just like regular scalar multiplication. So you can kind of foil this out, if t... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's just not normal multiplication. But we saw two videos ago that the dot product has a distributive and the associative and the commutative properties, just like regular scalar multiplication. So you can kind of foil this out, if that's how you remember multiplying your binomials. Or I always think of it more as jus... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Or I always think of it more as just doing the distributive property twice. So this can be rewritten as x dot x. Actually, let me write it as a distributive property, because that's sometimes not obvious to a lot of people. So let me write this x as a yellow x. And let me write this whole term as x plus y. So this righ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
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