In the last couple of videos, I already exposed you to the idea of a matrix, which is really just an array of numbers, and it's usually a two-dimensional array. Actually, it's always a two-dimensional array for our purposes. So if I have an m by n matrix, the m is just the number of rows, and then the n is just the number of columns. So let me write out the m by n matrix.
So I'll just specify, let's say I have the m by n matrix A. It's a capital bold A. And it's equal to, I'll be as general as possible, first entry is in, I'll just call that lowercase a, it's in row 1, column 1. The next entry is row 1, column 2. And you go all the way to row 1, column n.
You have n columns. And then when you go down. You go to the next row, will be row 2, column 1. And then you keep going all the way down to row m, column n, and then of course, what this entry's going to be, row 2, let me write that a little smaller, row 2, column 2. And you go all the way, and you're going to have row m, column n.
And so if you think about it, you're going to have how many total entries here? You're going to have m entries this way, n that way. So you're going to have m times n.
Total entries. And I think you're pretty familiar with this idea already of a matrix. You probably saw this in your Algebra 2 classes. So what we want to do now in this video is relate our notion of a matrix to everything we already know about vectors. Or maybe introduce some operations that allow matrix and vectors to interact with each other.
And maybe the most natural one is multiplication, or taking the product. So what I'm going to do in this video is define. I'm going to define what it means when we take the product of our matrix A, of any matrix A, I've written this as general as possible, with some vector x.
And our definition will only work, is only defined if x, if the vector we're multiplying a by, has the same number of components as a has columns. So this is only valid for an x that looks like this. x1, x2, all the way down to xn. So let me be very clear on this.
This vector right here, it could be a different height than this vector. What matters is that the same number of a's you have in this direction, you have na's here, then you have n components of this vector right here. And if you have that constraint, If the length of your vector, the number of components in vector, is equal to the number of columns in your matrix, then we define this product to be equal to, so that's my vector x.
So this is a definition. There's nothing in nature that told us it had to be defined this way. It's just human beings or mathematicians decided that this is a useful convention to define the multiplication or the product of a matrix and a vector.
So we're going to define. a times our vector x. These are both bold, or this is a matrix, that's a vector.
And the convention, if I didn't draw the little vector symbol in your textbooks, it'll just bold out the x. So it'll be a lowercase x. Lowercase is vector, uppercase is matrix. Both of them are bolded.
That tells you that you're not just dealing with regular numbers. So we're defining this to be equal to, let me write it out, fairly large. You're going to take each row.
You're going to take each one. We're going to show you that there's multiple ways to kind of visualize this. But it's going to be a11 times x1.
Let me write that down. So a11 times x1 plus a12. So a11 times x1 plus a12 times x2 all the way to plus a1n times xn. So that's just the first. So the product of this matrix, this m by n matrix, and this n-component vector, will be a new vector, the first entry of which is essentially each of these entries times the corresponding entry here, and you add them all up.
And as you can see, that's already looking fairly similar to a dot product. I'll discuss that in a second, but let me finish my definition before I start talking about what it means or what it might be related to. So that was that first row right there.
Like that. We just multiply that times this thing to get that row there. Now the second row, I'll do it in this, I wanted to do it in a different color. The second row, remember this is a definition.
Human beings came up with this. Nothing about nature said we had to do it this way, but it's just nice and convenient. So our second row, we'll have a21 times x1. We'll just do the whole thing over again, but this time we're multiplying this row times this column vector. So a21 times x1.
Plus a22 times x2, all the way until we get to a2n times xn. So we multiplied this entire row times that entire column, this term times that term, plus this term, plus this term, all the way down to this last term times that last. All the way down to plus this last term times that last term.
And then we keep doing this for every row until we get to the mth row. And then the mth row will be a m 1. Sorry, this is a m 1, right? This is the mth row first column.
a m 1 times x 1 plus, it's hard to keep switching colors, plus a m 2. times x2, all the way until we get to amn times xn. All the way until we get to amn times xn. So what is this vector going to look like? It's essentially going to have, if we called this vector, let's say we call this vector, let's say it's equal to vector b, what does vector b look like? How many entries is it going to have?
Well, it has an entry for each row of this, right? We're taking each row and we're multiplying it. We're essentially taking the dot product of this row vector with this column vector.
And I'll be a little bit more formal with the notation in a second, but I think you understand that this is a dot product. We're taking each corresponding, the first component times the first component, plus the second component times the second component, plus the third component times the third component, all the way to the nth component, plus the nth component times the nth component. So this is essentially the dot product of this row vector.
Row vector is just a fan, it's just you're writing a row, we've been writing all of our vectors as columns, so we could call them column vectors. You're just writing them as rows. And we can be a little bit more specific with the notation in a second.
But what's this going to look like? Well, we're doing this m times. So we're going to have m entries. You're going to have b1, b2, all the way to bn. If you view these as all as matrices, you can kind of view it as you have a You have, and this will eventually work for the matrix math we're going to learn.
This is an m by n matrix. And we're multiplying it by, how many rows does this guy have? He has n rows, right?
He has n components, and he has one column. So m by n times an n by 1, you essentially, you can kind of ignore these middle two terms, and they will result with, how many rows does this guy have? He has m rows and one column.
So if you take out these The middle two terms have to be equal to each other just for the multiplication to be defined. And then you're left with an m by 1 matrix. So this was all abstract.
Let me actually apply it to some actual numbers. But it's important to actually set the definition. Now that we have the definition, we can apply it to some actual matrices and vectors.
So let's say we have the matrix. Let's say I want to multiply the matrix. Minus 3, 0, 3, 2. I'll do this one in yellow.
1, 7, minus 1, 9. And I want to multiply that by the vector. Now how many components or rows does this vector have to have? Well, my matrix times vector product, or multiplication, is only defined if my vector has as many components as this matrix has columns. So we have 1, 2, 3, 4 columns, so this guy's got to have 4 components, first to even be able to multiply them. Otherwise it wouldn't be defined.
So let me put 4 entries here. Let's say it's 2, minus 3, 4, and then minus 1. So what is this going to be equal to? This is equal to, we just, we essentially take the dot. The first term of this is going to be the dot product of this first row.
with this vector. And then the second entry here is going to be the dot product of this row vector with this column. So let's do it. So it's going to be minus 3 times 2. I'm not going to color code it.
Minus 3 times 2. Plus 0 times minus 3 plus 3 times 4 plus 2 times minus 1. And now my second row, or I guess my second component in this vector, is going to be 1 times 2. So 1 times 2 plus 7 times negative 3 plus minus 1 times 4 minus 1. times 4 plus 9 times minus 1. And so what is this simplified to? This is equal to minus 3 times 2 is minus 6 plus 0 plus 12, minus 2. And then this is simplified to 2 minus 21 minus 4. Let me see, that looked like a minus. Minus 21 minus 4 minus 9. So this is equal to this top term. Let's see, I have a minus 6 plus 12 is 6, minus 2 is 4. And then I have 2 minus 21 is minus 19. I want to make sure I get the math right here. Let me see.
Minus 21 minus 9 is minus 30. And I have a minus 34. And then I have a plus 2, so minus 32. So that's my product right there. And like I said, you can view it, and let me be very clear right here. Well, everything we've been used to right now, we've been writing our vectors as column vectors.
But you can view each of these guys right here as a row vector, but let me be even better. Let's say that vector, let me call vector a, A1. So let me define vector A1 is equal to minus 3, 0, 3, 2. And let me define vector A2 to be equal to 1, 7, minus 1, 9. So all I did is I wrote these guys, but I wrote them in our standard vector form.
I wrote them as column vectors. So what we can write, what we can define, To turn these guys into row vectors is the transpose function. And transpose, you just turn the rows into columns and the columns into rows. So if this is a1, then a1 transpose will just be the row version of this. So it's minus 3, 0, 3, 2. And then a2 transpose would be equal to 1, 7, minus 1, And 9. And then this multiplication right here, we can rewrite it as, this matrix, we can rewrite it as, we have vector a1 transpose for the first row.
a1 transpose, that's a vector. These are vectors. Now, row vectors. And then this is a2 transpose. Transpose should be the superscript.
This vector can be written exactly like this, right? Because this is the first row, this is the second row. Times the vector times, let me just call this vector x.
That right there is vector x. So this is right here, vector x. We can now rewrite the definition as this would be equal to what?
This first row right here. that we wrote out. The first row that we wrote out right here.
This was a1 dot x. And you know all about the dot products. The first row was a1 dot x. It's minus 3 times 2 plus 0 times minus 3 plus 3 times 4. It's a1 dot x.
And this is useful because when I defined the dot product, I only defined it with column vectors like this. So now I'm dotting two column vectors. I haven't formally defined a row vector times a column vector. So now I can say, look, if this is just a standard column vector like we've been working with, I can write my matrix as each row is the transpose of a column vector, or it's a row vector, then I can write this product as just the dot products of each of these transpose, or I guess you could say the inverse transpose. with this vector right here.
And then obviously the second row is going to be a2, vector vector a2 dot x. The second row is a2 dot x is 1 times 2 plus 7 times minus 3, right there, minus 1 times 4, plus 9 times minus 1. So just like that. So this is one way to view it.
It's kind of matrix times a vector is just like the transpose of its rows dotted. with the vector you're multiplying it by. This is one way to perceive matrix multiplication. Now the other way to perceive it, let me do it this way. Well, I'll do it with a different example.
Those numbers are getting a little bit tiresome. Let's say I have the matrix A, nice and bold, is equal to, oh, I don't know. Let me just say it's. 3, 1, 0, 3, 2, 4, 7, 0, minus 1, 2, 3, and 4. And I need to multiply this.
I have to multiply this times a four-component vector. So let me call vector x is equal to 5. Actually, let me just keep it general. Let's say it's equal to x1, x2, x3, and x4. Now, instead of viewing these as row vectors, we could view A as a set of column vectors.
We could call this thing right here, we could call that vector 1. We call this thing right here vector 2. We call this thing right here vector 3. And we call this thing right here vector 4. Then we could rewrite our matrix A as being equal to Just a bunch of column vectors. So we could rewrite it as vector 1, vector 2, vector 3, vector 1, and vector 4. That's vector 4 right there. So how can the matrix multiplication be interpreted in this context? Well, what did we do?
We multiplied. When we multiply these guys, we always do, all of the elements in here always get multiplied by x1, right? It's 3, let me start some of the multiplication here, just from our definition. So if I multiply a times x, I'll start it off. Maybe I won't do the whole thing.
I just want you to see the pattern. It's 3 times x1 plus 1 times x2 plus 0 times x3 plus 3 times x4, that's the first entry. And then you have 2 times x1 plus 4. times x2 all the way, and then you finally have minus 1 times x1 plus 2 times x2, you get the idea.
But what's happening here? This guy, this first vector, is always being multiplied by the scalar x1. In fact, you can view this part of the entries right here, right?
We're just multiplying this guy times the scalar of x1. In every case. Right? You have 3, 2, minus 1, 3, 2, minus 1. We're multiplying by the scalar of x1.
And then we're adding that to this guy times the scalar x2. Right? And then we're adding that to this guy times the scalar x3.
So we can rewrite a times x as being equal to the scalar x1 The scalar x1 times the vector v1 plus the scalar x2. This is the scalar x1 times the vector v1 plus the scalar x2 times the vector v2. I want to do that in yellow.
Times the vector v2 plus x3 times the vector v3. Plus the scalar x4 times the vector v4. And obviously, if we had n terms here, we'd have to have n vectors here.
And we could just make this more general to n. But what's interesting here is now the product ax, it can be interpreted as a linear combination. These are just arbitrary numbers depending on what our vector x is. So depending on our vector x, we're taking a linear combination of the column vectors of A.
So this is a linear combination of column vectors of A. So this is really interesting. I'm sure you've been exposed to matrix multiplication in the past, but I really want you to absorb these two ways of interpreting it, because they'll be important when we talk about column spaces. and things like that in the future, is that you can interpret matrix multiplication. And actually, there's other ways you can interpret it as a transformation of the spectra of x, but I won't cover that in this video just for brevity.
But you can interpret it as a weighted combination or linear combination of the column vectors of A, where the matrix x dictates what the weights on each of the columns are. Or you can interpret it as essentially the dot product of the row vectors, or you could define the row vectors as the transpose of column vectors, the dot product of those column vectors, each of the corresponding column vectors with your matrix X. So these are both completely valid interpretations. And hopefully, at least this video, at least gives you a working knowledge of what matrix multiplication is. And even better, it gives you a little bit deeper sense of all of the different ways that it can be interpreted.