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And B is a 3 by 4 matrix. So by our definition, what is the product AB going to be equal to? Well, we know it's well-defined because the number of columns here is equal to the number of rows. So we can actually take these matrix-vector products. You'll see that in a second. So AB is equal to the matrix A times the colu... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So we can actually take these matrix-vector products. You'll see that in a second. So AB is equal to the matrix A times the column vector 1, 2, 3. That's going to be the first column in our product matrix. And the second one is going to be the matrix A times the column 0, 0, 1. The third column is going to be the matri... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
That's going to be the first column in our product matrix. And the second one is going to be the matrix A times the column 0, 0, 1. The third column is going to be the matrix A times the column vector 1, 1, 0. And then the fourth column in our product vector is going to be the matrix A times the column vector 1, minus ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And then the fourth column in our product vector is going to be the matrix A times the column vector 1, minus 1, 2. And when we write it like this, it should be clear why this has to be why the number of columns in A have to be the number of rows in B. Because the column vectors in B are going to have the same number o... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So all of the column vectors in B, so if we call this B1, B2, B3, B4, all of my Bi's, let me write it this way, all of my Bi's, where this i could be 1, 2, 3, or 4, are all members of R3. So we only have matrix vector products well defined when the number of columns in your matrix are equivalent to essentially the dime... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, now we've reduced our matrix matrix product problem to just a bunch of four different matrix vector product problems, so we can just multiply these. This is nothing new to us. So let's do it. And so what is this equal to? So AB, let me rewrite it, AB, my product vector, is going to be equal to, so this first colu... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And so what is this equal to? So AB, let me rewrite it, AB, my product vector, is going to be equal to, so this first column is the matrix A times the column vector 1, 2, 3. And how did we define that? Remember, one way to think about it is that this is equal to, you can kind of think of it as each of the rows of A dot... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Remember, one way to think about it is that this is equal to, you can kind of think of it as each of the rows of A dotted with the column here of B. Or even better, this is the transpose of some matrix. Like let me write this this way. If A is equal to, sorry, the transpose of some vector. Let's say that A is equal to ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
If A is equal to, sorry, the transpose of some vector. Let's say that A is equal to the column vector 0, minus 1, 2, then A transpose, and I haven't talked about transposes a lot yet, but I think you get the idea. You just change all of the columns into rows. So A transpose will just be equal to 0, minus 1, 2. You just... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So A transpose will just be equal to 0, minus 1, 2. You just go from a column vector to a row vector. So if we call this thing here A transpose, then when we take the product of our matrix A times this vector, we essentially are just taking A and dotting with this guy for our first row and our first column. So let me d... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me do it that way. So let me write in that notation. So this is going to be the vector 1, minus 1, 2. That's essentially that row right there represented as a column dotted with 1, 2, 3. Actually, let me do it in that color. Just so I can later switch to one color to make things simple, but dotted with 1, 2, 3. ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
That's essentially that row right there represented as a column dotted with 1, 2, 3. Actually, let me do it in that color. Just so I can later switch to one color to make things simple, but dotted with 1, 2, 3. So we just took that row, or I guess the column equivalent of that row, and dotted with this. And I wrote it ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So we just took that row, or I guess the column equivalent of that row, and dotted with this. And I wrote it like this because we've only defined dot products for column vectors. I could do it maybe for row vectors, but who would need to make a new definition? So that's going to be the first entry in this matrix vector... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So that's going to be the first entry in this matrix vector product. The second entry is going to be the second row of A, essentially dotted with this vector right there. So it's going to be equal to 0, minus 2, and 1, dotted with 1, 2, 3. Dotted with 1, 2, 3. And we just keep doing that. And I'll just switch maybe to ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Dotted with 1, 2, 3. And we just keep doing that. And I'll just switch maybe to one neutral color now. So then A times 0, 0, 1. That's going to be the first row of A expressed as a column vector. So we can write it like this. 1, minus 1, 2, dot 0, 0, 1. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So then A times 0, 0, 1. That's going to be the first row of A expressed as a column vector. So we can write it like this. 1, minus 1, 2, dot 0, 0, 1. And then, actually, and then we have our, and then the second row of A dotted with this column vector. So we have 0, minus 2, 1, dotted with 0, 0, 1. Two more rows left. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
1, minus 1, 2, dot 0, 0, 1. And then, actually, and then we have our, and then the second row of A dotted with this column vector. So we have 0, minus 2, 1, dotted with 0, 0, 1. Two more rows left. This can get a little tedious. And it's inevitable that I'll probably make a careless mistake, but as long as you understa... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Two more rows left. This can get a little tedious. And it's inevitable that I'll probably make a careless mistake, but as long as you understand the process, that's the important thing. So the next one, this row of A expressed as a column vector. 1, minus 1, 2, we're going to dot it with this vector right there. 1, 1, ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So the next one, this row of A expressed as a column vector. 1, minus 1, 2, we're going to dot it with this vector right there. 1, 1, 0. And then this row of A, I could just look over here as well, 0, minus 2, 1, dotted with 1, 1, 0. And then finally, the last two entries are going to be the top row of A. 1, minus 1, 2... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And then this row of A, I could just look over here as well, 0, minus 2, 1, dotted with 1, 1, 0. And then finally, the last two entries are going to be the top row of A. 1, minus 1, 2, dotted with this column vector, 1, minus 1, 2, put a little dot there. Remember, we're taking the dot product. And then finally, this s... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Remember, we're taking the dot product. And then finally, this second row of A. So 0, minus 2, 1, dotted with this column vector. 1, minus 1, 2. And that is going to be our product matrix. And this looks very complicated right now, but now we just have to compute in dot products, tend to simplify things a good bit. So ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
1, minus 1, 2. And that is going to be our product matrix. And this looks very complicated right now, but now we just have to compute in dot products, tend to simplify things a good bit. So what is our matrix, our product, going to simplify to? I'll do it in pink. AB is equal to, let me draw the matrix right there. So ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So what is our matrix, our product, going to simplify to? I'll do it in pink. AB is equal to, let me draw the matrix right there. So what's the dot product of these two things? It's 1 times 1. I'll just write it out. It's 1 times 1. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So what's the dot product of these two things? It's 1 times 1. I'll just write it out. It's 1 times 1. I'll just write 1 times 1 is 1, plus minus 1 times 2, so minus 2, plus 2 times 3, plus 6. Now we'll do this term right here. 0 times 1 is 0, plus minus 2 times 2, so that's minus 4, plus 1 times 3, plus 3. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
It's 1 times 1. I'll just write 1 times 1 is 1, plus minus 1 times 2, so minus 2, plus 2 times 3, plus 6. Now we'll do this term right here. 0 times 1 is 0, plus minus 2 times 2, so that's minus 4, plus 1 times 3, plus 3. Now we're on to this term. 1 times 0 is 0, plus minus 1 times 0, plus 0, plus 2 times 1, is equal ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
0 times 1 is 0, plus minus 2 times 2, so that's minus 4, plus 1 times 3, plus 3. Now we're on to this term. 1 times 0 is 0, plus minus 1 times 0, plus 0, plus 2 times 1, is equal to plus 2. This term, 0 times 0 is 0, plus minus 2 times 0, let me write it as 0, plus minus 2 times 0 is 0, plus 1 times 1. So plus 1. Then ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
This term, 0 times 0 is 0, plus minus 2 times 0, let me write it as 0, plus minus 2 times 0 is 0, plus 1 times 1. So plus 1. Then here you have 1 times 1 is 1, plus minus 1 times 1 is minus 1, plus 2 times 0, so plus 0. Here, 0 times 1 is 0, minus 2 times 1 is minus 2, and then 1 times 0 is plus 0. Almost done. 1 times... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Here, 0 times 1 is 0, minus 2 times 1 is minus 2, and then 1 times 0 is plus 0. Almost done. 1 times 1 is 1, minus 1 times minus 1 is 1, 2 times 2 is 4. Finally, 0 times 1 is 0, minus 2 times minus 1 is 2. 1 times 2 is also 2. And we're in the home stretch. Now we just have to add up these values. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Finally, 0 times 1 is 0, minus 2 times minus 1 is 2. 1 times 2 is also 2. And we're in the home stretch. Now we just have to add up these values. So our dot product of the two matrices is equal to the 2 by 4 matrix, 1 minus 2 plus 6, that's equal to 5, minus 4 plus 3 is minus 1. This is just 2, this is just 1. Then we ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Now we just have to add up these values. So our dot product of the two matrices is equal to the 2 by 4 matrix, 1 minus 2 plus 6, that's equal to 5, minus 4 plus 3 is minus 1. This is just 2, this is just 1. Then we have 1 minus 1 plus 0 is just 0, minus 2, we just have a minus 2 there, 1 plus 1 plus 4 is 6, and then 2 ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Then we have 1 minus 1 plus 0 is just 0, minus 2, we just have a minus 2 there, 1 plus 1 plus 4 is 6, and then 2 plus 2 is 4, and we are done. The product of AB is equal to this matrix right here. Let me get my A and B back. We can talk a little bit more about what this product actually represented. So let me copy and ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
We can talk a little bit more about what this product actually represented. So let me copy and paste this. And then I'll paste it. Let me scroll down a little bit. Go down here, paste. There you go. So this was our A and our B. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me scroll down a little bit. Go down here, paste. There you go. So this was our A and our B. And when we took the product, we got this matrix here. Now there are a couple of interesting things to notice. Remember, I said that this product is only well defined when the number of columns in A is equal to the number o... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So this was our A and our B. And when we took the product, we got this matrix here. Now there are a couple of interesting things to notice. Remember, I said that this product is only well defined when the number of columns in A is equal to the number of rows in B. So that was the case in this situation. And then notice... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Remember, I said that this product is only well defined when the number of columns in A is equal to the number of rows in B. So that was the case in this situation. And then notice, we got a 2 by 4 matrix, which is the number of rows in A times the number of columns in B. So we got a 2 by 4 matrix. So another natural q... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So we got a 2 by 4 matrix. So another natural question is, could we have found, or is it even equal, if we were to take the product BA? So if we tried to apply our definition there, what would it be equal to? It would be equal to the matrix B times the column 1, 0. Then the matrix B times the column minus 1, minus 2. A... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
It would be equal to the matrix B times the column 1, 0. Then the matrix B times the column minus 1, minus 2. And then it would be the matrix B times the column 2, 1. Now, can we take this matrix vector product? We have a 3 by 4. This right here is a 3 by 4 matrix. And this guy right here is a member of R2. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Now, can we take this matrix vector product? We have a 3 by 4. This right here is a 3 by 4 matrix. And this guy right here is a member of R2. So this is not well defined. We have more columns here than entries here. So we have never defined a matrix vector product like this. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And this guy right here is a member of R2. So this is not well defined. We have more columns here than entries here. So we have never defined a matrix vector product like this. So not only is this not equal to this, it's not even defined. So it's not defined when you take a 3 by 4 matrix and you take the product of tha... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So we have never defined a matrix vector product like this. So not only is this not equal to this, it's not even defined. So it's not defined when you take a 3 by 4 matrix and you take the product of that with a 2 by 4 matrix. It's not defined because that number and that number is not equal. And so obviously, since th... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
It's not defined because that number and that number is not equal. And so obviously, since this is defined and this isn't defined, you know that AB is not always equal to BA. In fact, it's not usually equal to BA. And sometimes it's not even defined. And the last point I want to make is you probably learned to do matri... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And sometimes it's not even defined. And the last point I want to make is you probably learned to do matrix-matrix products in Algebra 2, but you didn't have any motivation for what you were doing. But now we do have a motivation. Because when you're taking the product of A and B, we learned in the last video that if w... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
Because when you're taking the product of A and B, we learned in the last video that if we have two transformations, we have two transformations. Let's say we have the transformation S is a transformation from R3 to R2. And that S is represented by the matrix. So S, given some matrix in R3, if you apply the transformat... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So S, given some matrix in R3, if you apply the transformation S to it, it's equivalent to multiplying that, or given any vector in R3, applying the transformation S is equivalent to multiplying that vector times A. We can say that. And I used R3 and R2 because the number of columns in A is 3. So it can apply to a thre... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So it can apply to a three-dimensional vector. And similarly, we can imagine B as being the matrix transformation of some transformation T that is a mapping from R4 to R3, where if you give it some vector x in R4, it will produce, you take the product of that with B, and you're going to get some vector in R3. Now, if w... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
If we have R4 here, let me switch colors. We have R4 here. We have R3 here. And then we have R2 here. T is a transformation from R4 to R3. So T would look like that. T is a transformation. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And then we have R2 here. T is a transformation from R4 to R3. So T would look like that. T is a transformation. It's B times x. That's what T is equal to. So T is this transformation. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
T is a transformation. It's B times x. That's what T is equal to. So T is this transformation. And then S is a transformation from R3 to R2. So S looks like that. And S is equivalent to A times any vector in R3. | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So T is this transformation. And then S is a transformation from R3 to R2. So S looks like that. And S is equivalent to A times any vector in R3. So that is S. So now we know how to visualize or how to think about what the product of A and B are. The product of A and B is essentially, you apply the transformation B fir... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
And S is equivalent to A times any vector in R3. So that is S. So now we know how to visualize or how to think about what the product of A and B are. The product of A and B is essentially, you apply the transformation B first. So let me think of the composition of S. Let me write it this way. So what is the composition... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me think of the composition of S. Let me write it this way. So what is the composition of S with T? This is equal to S of T of x. So you take a transformation from R4 to R3, and then you take the S transformation from R3 to R2. So this is S of T. S of T is a transformation from R4 all the way to R2. And then the... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So you take a transformation from R4 to R3, and then you take the S transformation from R3 to R2. So this is S of T. S of T is a transformation from R4 all the way to R2. And then the neat thing about this, if you were to just write this out in its matrix representations, we did this in the last video, this would be eq... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
But now we know that the matrix, by our definition of matrix vector products, that this guy right here is going to have a transformation. It's going to be equal to, so the composition S of T of x is going to be equal to the matrix AB, based on our definition, so the transformation AB times some vector x. So the reason ... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
We took the pain of multiplying the matrix A times the matrix B, and we got this value here. And hopefully I didn't make any careless mistakes. But the big idea here, the idea that you probably weren't exposed to in your Algebra 2 class, is that this is the matrix of the composition of the transformations S and T. So r... | Matrix product examples Matrix transformations Linear Algebra Khan Academy.mp3 |
So we could draw it right over here. So it's equal to two, one. So the horizontal, if we were to start at the origin, we would move two in the horizontal direction and one in the vertical direction. So we would end up right over here. Now what I wanna do is think about how we can define multiplying this vector by a sca... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we would end up right over here. Now what I wanna do is think about how we can define multiplying this vector by a scalar. So for example, if I were to say, if I were to say three times, three times the vector a, which is the same thing as saying three times two, one. So three is just a number. One way to think abou... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So three is just a number. One way to think about a scalar quantity, it is just a number versus a vector. This is giving you, it's giving you how much you're moving in the various directions right over here. It's giving you both a magnitude, magnitude and a direction, while this is just a plain number right over here. ... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's giving you both a magnitude, magnitude and a direction, while this is just a plain number right over here. But how would we define multiplying three times this vector right over here? Well, one reasonable thing that might jump out at you is well, why don't we just multiply the three times each of these components?... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this could be equal to, so we have two and one, and we're gonna multiply each of these times three. So three times two and three times one, and then the resulting vector is still going to be a two-dimensional vector, and it's going to be the two-dimensional vector six, three. Now I encourage you to get some graph pa... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me do that. So the vector six, three, if we started at the origin, we would move six in the horizontal direction, one, two, three, four, five, six, and three in the vertical, one, two, three. So it gets us right over there. So it would look like this. So what just happened to this vector? Well, notice, one way t... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it would look like this. So what just happened to this vector? Well, notice, one way to think about it is what's changed and what has not changed about this vector? Well, what's not changed is still pointing in the same direction. So this right over here has the same direction. Multiplying by the scalar, at least th... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, what's not changed is still pointing in the same direction. So this right over here has the same direction. Multiplying by the scalar, at least the way we defined it, did not change the direction that my vector is going in, or at least in this case it didn't, but it did change its magnitude. Its magnitude is now ... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Its magnitude is now three times longer, which makes sense, because we multiplied it by three. One way to think about it is we scaled it up by three. The scalar scaled up the vector. That might make sense, or it might give an intuition of where that word scalar came from. The scalars, when you multiply it, it scales up... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
That might make sense, or it might give an intuition of where that word scalar came from. The scalars, when you multiply it, it scales up a vector. It increased its magnitude by three without changing its direction. But let's do something interesting. Let's multiply our vector A. Let's now multiply it by a negative num... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
But let's do something interesting. Let's multiply our vector A. Let's now multiply it by a negative number. Let's actually just multiply it by negative one, just for simplicity. So let's just multiply negative one times A. Well, using the convention that we just came up with, we would multiply each of the components b... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's actually just multiply it by negative one, just for simplicity. So let's just multiply negative one times A. Well, using the convention that we just came up with, we would multiply each of the components by negative one. So two times negative one is negative two, and one times negative one is negative one. So now... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So two times negative one is negative two, and one times negative one is negative one. So now negative one times A is going to be negative two, negative one. So if we started at the origin, we would move in the horizontal direction, negative two, and in the vertical direction, negative one. So now what happened to the ... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So now what happened to the vector? Now what happened to the vector? When I did that, well, now it flipped its direction. Multiplying it by this negative one, it flipped its direction. Its magnitude actually has not changed, but its direction is now in the exact opposite direction, which makes sense that multiplying by... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Multiplying it by this negative one, it flipped its direction. Its magnitude actually has not changed, but its direction is now in the exact opposite direction, which makes sense that multiplying by a negative number would do that. In fact, when we just dealt with the traditional number line, that's what happened. If y... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
If you took five times negative one, well, now you're going in the other direction. You're at negative five. You're five to the left of zero. So it makes sense that this would flip its direction. So you could imagine if you were to take something like negative two times your vector A, negative two times your vector A, ... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it makes sense that this would flip its direction. So you could imagine if you were to take something like negative two times your vector A, negative two times your vector A, and I encourage you to pause this video and try this on your own, what would this give and what would be the resulting visualization of the ve... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
This would be equal to negative two times two is negative four negative two times one is negative two. So this vector, if you were to start at the origin, remember, you don't have to start at the origin, but if you were, it would be, so you'd go zero, one, two, three, four, one, two. It looks just like this. It looks j... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
It looks just like this. And so just to remind ourselves, our original vector A looked like this. Our original vector A looked like this. Two, one looks like this. And then when you multiply it by negative two, you get a vector that looks like this. You get a vector that looks like this. Let me draw it like this. | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Two, one looks like this. And then when you multiply it by negative two, you get a vector that looks like this. You get a vector that looks like this. Let me draw it like this. And purposely not having them all start at the origin because they don't have to all start at the origin, but you get a vector that looks like ... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me draw it like this. And purposely not having them all start at the origin because they don't have to all start at the origin, but you get a vector that looks like this. That looks like this. So what's the difference between A and negative two times A? Well, the negative flipped it over and then the two flipped it... | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
So what's the difference between A and negative two times A? Well, the negative flipped it over and then the two flipped it over and now it has twice the magnitude, but because of the negative, it has twice the magnitude in the other direction. | Multiplying a vector by a scalar Vectors and spaces Linear Algebra Khan Academy.mp3 |
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