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- In the last couple of videos, I already exposed you to the
- idea of a matrix, which is really just an array of
- numbers, usually a 2-dimensional array.
- Actually it's always a 2-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 have the m by n matrix A, it's
- a capital bold A.
- And it is 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, it 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 is 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 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 II 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 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 if x, 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 x n.
- So let me be very clear with this, this vector, I guess you
- could do it a different height than this vector.
- What matters is that the same number of A's you have in this
- direction, you have n A's here, then you have n
- components of this vector right here.
- And if you have that constraint, if the length of
- your vector, or 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 this is 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 the define the multiplication,
- or the product, of a matrix and a vector.
- So we'll define A times our vector x.
- These are both bold, this is a matrix, that's a vector.
- And the convention, if I didn't draw the little vector
- symbol, your textbooks would just bold out the x, so that
- it'll be a lowercase x.
- Lower case 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, and we're going to show you
- that there's multiple ways to kind of visualize this, but
- it's going to a11 times x1, let me write that down.
- So a11 times x1 plus a12 times x2, all the way to
- plus a1n times xn.
- 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 a
- corresponding entry here, and you add them all up.
- And as you can see, that's already looking fairly similar
- to a dot product, and 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, it'll
- just look like that.
- We just multiply that times this thing to
- get that row there.
- Now the second row -- I want to do it in a different color
- -- 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 will 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
- -- I wanted to do that in magenta -- 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 plus this last term
- times that last term.
- And we keep doing this for every row until we get to the
- m-th row, and then the m-th row will be am1.
- This is the m-th row first column.
- am1 times x1 plus -- it's hard to keep switching colors --
- plus am2 times x2, 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 -- 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 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.
- 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 n-th component plus the n-th component
- times the n-th component.
- So this is essentially the dot product of this row vector.
- 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 look like?
- Well we're doing this m times, so we're
- going to have m entries.
- You're going to b1 b2 all the way to bn.
- If you viewed these all as matrices, you can kind of view
- it as -- 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.
- He has n components, and he has 1 column.
- So m by n times an n by 1, you essentially can ignore these
- middle two terms, and they'll result with -- how many rows
- does this guy have?
- He has m rows, and 1 column.
- These 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.
- Now 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 going to have 4 components for us even to 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?
- The first term of this is going to be the dot product of
- this first row with this vector.
- And then the second entry 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 plus 7 times negative
- 3 plus minus 1 times 4 plus 9 times minus 1.
- And so what does this simplify to?
- This is equal to minus 3 times 2 is minus 6 plus 0 plus 12.
- This is 12.
- Minus 2.
- And then this is simplified to 2 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.
- 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 let me be very clear right here.
- Everything we've been used to right now, we've been writing
- our vectors as column vectors.
- But you can view each of these 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 define to turn these guys into row vectors is
- the transpose function.
- In 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 -- we have vector a1 transpose for the first row.
- These are vectors now, row vectors.
- And then this is a2 transpose.
- The transpose should be the super script.
- This vector can be written exactly like this because this
- is the first row, this is the second row.
- Times the vector, let me just call this vector x, that right
- there is vector x.
- We can now rewrite the definition as this would be
- equal to what?
- This first row right here that we wrote out,
- this was a1 dot x.
- 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.
- And I'm dotting 2 column vectors.
- I haven't formally defined a row vector
- times a column vector.
- So now I can say if this is just a standard column factor,
- 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 dot x.
- The second row is a2 dot x, is 1 times 2 plus 7 times minus 3
- minus 1 times 4 plus 9 times minus 1.
- So just like that.
- So this is one way to view it.
- Matrix times the vector is just like the transpose of its
- rows dotted with the vector you're ds it by.
- This is one way to perceive matrix multiplication.
- Now the other way to perceive it -- let me 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 3,
- 1, 0, 3, 2, 4, 7, 0, minus 1, 2, 3, and 4.
- And I need to multiply this times a 4 component vector.
- So let me call vector x is 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 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 vector 1, vector 2, vector
- 3, and vector 4.
- So how can the matrix multiplication be interpreted
- in this context?
- Well what did we do?
- When we multiply these guys, all of the elements in here
- always get multiplied by x1.
- 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 first vector is always being multiplied
- by the scalar x1.
- In fact you can view this part of the entries right here.
- We're just multiplying this guy times the scalar of x1 in
- every case.
- 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
- 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
- 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.
- 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 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.
- Actually there's other ways you can actually interpret
- that as a transformation of this vector x.
- But I won't cover that in this video just for brevity.
- But you can interpret it as a weighted combination, or a
- 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 a 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 this video at least gives you a working knowledge
- of what matrix multiplication is.
- And even better, gives you a little bit deeper sense of all
- of the different ways that it can be interpreted.