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Current time:0:00Total duration:10:41

In the last video we started
off with two linear transformations. We had the linear transformation
s that was a mapping from Rn to Rm. And then we had the linear
transformation t, that was also a mapping from Rn to Rm. And we defined the idea of
the addition of these two transformations. So s plus t, this transformation
of x we defined as being equal to s of x,
this vector plus t of x. And of course, this input is
still from Rn, and then each of these are vectors in Rm. If we add two vectors in Rm to
each other, we get another vector in Rm because Rm
is a valid subspace. It's also closed
under addition. So this is still a mapping. So s plus t is still a mapping
from Rn to Rm. And we also said that every
linear transformation we've shown in a previous video, can
be represented as a matrix. We could say that s of
x is equal to some matrix a times x. And we could also say that
t of x is equal to some matrix b times x. And both of these would
be m by n matrices. And let me write that m
by n, both of these. Because these are both mappings
from Rn to Rm. And what we did is we made
a another definition. This was a definition right here
and then we made another definition. We defined the addition
of two matrices. We said any matrix a plus b,
they both have to have the same dimensions. So they're both m by
n in this case. And we defined this addition to
be a new matrix, where each column of this matrix is the
sum of the corresponding columns of these matrices. So this matrix's first column
will be the sum of a's first column and b's first column. So a1 plus b1, the second column
I'll do a little line here is, a2 plus b2. And it goes all the
way to An plus Bn. This was a definition. And the whole reason why we
made this definition, is because if you defined matrix
addition in this way, then this thing, when you replace it
with Ax plus Bx, you get to that this thing over here is
equal to the corresponding matrices by this definition,
a plus b times x. This was the motivation to get
to a nice expression, like this, for defining matrix
addition in this way. Now this all seems very
abstract, so let's actually add a matrix, or let's
add two matrices. So we'll start off with
a two-by-two case. So let's say I'm adding the
matrix 1, 3, minus 2, 4 to the matrix, remember they have to
have the same dimensions, to the matrix 2, 7, minus
3, minus 1. What do we get? Well by definition, you
just add up their corresponding columns. You add up the first column. When you add up the
corresponding columns, what happens when you add up two
columns, two vectors? Well, you just add up their
corresponding entries. So essentially, when you add
up to matrices, you're just adding up all of the
corresponding entries. I'll talk about it in this way,
just because that's how I defined it, but they're
all equivalent. The first thing, the first
column, in this matrix right here, is going to be this
column plus this column. So it's going to be 1 plus 2,
let me write it like this, and then minus 2, minus 3. And then the second column, that
one right there, is going to be 3 plus 7 and
then 4 minus 1. And so this will be equal
to 3, 10, minus 5, and 3, just like that. And notice, even though the
definition is I'm adding up corresponding columns. Well, what in effect happened? Well, I'm just adding up the
corresponding entries. I added the 1 to the 2, the 3
to the 7, the minus 2 to the minus 3, the 4 to the minus 1. It's that straightforward. Nothing fancier than that. In fact, we could have rewritten
this definition. If we say that the vector or the
matrix a is equal to a11 a12, all the way to a1n. And then if you go down this
is a21 all the way to a1n. And then you go all the
way down there to ann. We've seen that before. And then the matrix b, just
by the same argument or by similar definition, this is b11,
that entry is b11, that's b12, all the way to b1n. This is b21, second row, all the
way down to bn, sorry this is m, we have m rows,
so this is mn. So this right here is bm1, this
would be bm2, all the way down to this is bmn,
right there. Want to be very careful, these
are m by n matrices. So this row down here is the
mth row in both cases. But we could redefine our
matrix, or another way of stating this matrix addition
definition, is to say, if I'm going to add a plus b, I'm
just going to add up the corresponding entries. So this entry up here is going
to be-- do it in a different color --it's going to be a11
plus b11 this one's going to be a21 plus b21 all the way
down to am1 plus bm1. And then this is going to be,
of course, a12 plus b12 all the way to a1n-- let me scroll
over a little bit --all the way over to a1n plus b1n,
and then all the way down to amn plus bmn. These are equivalent
definitions. This takes a lot less space
to write in and I felt comfortable doing it because
we've already defined vector addition. But it essentially boils down
to you just add up all the corresponding entries. That's all matrix addition is. It's probably one of the
simplest things that you've seen in your recent mathematical
experience. Now, matrix scalar
multiplication, very similar idea. We defined scalar multiplication
times a transformation of x to be equal
to a scalar times the transformation of x. This was a definition. And we also defined scalar
times some matrix a to be equal to the scalar. A new matrix where each of its
columns are the scalar times the column vectors of a. So a1, and then the next column
is ca2, and then you go all the way to can. And the whole motivation for
this was, because this could be simplified to-- well t I've
said was equal to Bx, a times the transformation of x
--the transformation t of x was equal to. So we still have our c. So it's going to be c
times the matrix B, times the vector x. That's what the transformation
of x could be written as. And so this would be equal to by
just manipulating-- and we did this in the last video by
actually breaking this up in the column vectors multiplying
them by each of the components of x, and then distributing the
c and rearranging them a little bit. We can now say, using this
definition, that this is equal to some new matrix cB. We're using this definition,
some new matrix cB, where it's essentially c times
each of the column vectors of B times x. This right here was
our motivation. We wanted to be able express
this as a product of some new matrix and a vector, because
any linear transformation should be expressible
in that way. And that's why we made
this definition. Now let's apply it. And I wanted to show you that
this is perhaps even simpler than matrix addition. So if we want to multiply the
scalar 5 times the matrix, I'll do a 3 by 2 matrix. So 1, minus 1, 2, 3, 7, 0. This will just be equal to--
by this definition I'm just saying, I'm multiplying the
scalar times each of the column vectors. So it's 5 times 1, 2, 7. But what's that? That's just 5 times each
of the entries. It's 5 times 1, which is 5. 5 times 2, which is 10. 5 times 7, which is 35. Then the next column is going
to be 5 times this column right here, which is just 5
times each of its entries. So 5 times minus 1 is minus 5. 5 times 3 is 15. 5 times 0 is 0. It's that simple. You literally, if we go back
to this definition, we can define scalar multiplication
of a matrix. So we could also define cA as
being equal to, if we'd say this is a representation for A,
of the scalar c times each of the entries of A. That's it. So it's c times a11,
c times a12 all the way to c times a1n. You go down this way, c times
a21 all the way down to c times am1 and then if you
go down the diagonal, it's be c times amn. You literally just take your
scalar and multiply it times every entry in A. And that's all you have to do. So hopefully this clarified
things up a little bit, or maybe it was a bit of a review
from things you might have learned in highschool.