Main content

## Linear algebra

### Unit 2: Lesson 1

Functions and linear transformations- A more formal understanding of functions
- Vector transformations
- Linear transformations
- Visualizing linear transformations
- Matrix from visual representation of transformation
- Matrix vector products as linear transformations
- Linear transformations as matrix vector products
- Image of a subset under a transformation
- im(T): Image of a transformation
- Preimage of a set
- Preimage and kernel example
- Sums and scalar multiples of linear transformations
- More on matrix addition and scalar multiplication

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# Sums and scalar multiples of linear transformations

Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication. Created by Sal Khan.

## Video transcript

Let's say I have two
transformations. I have the transformation S,
which is a function or a transformation from Rn to
Rm, and I also have the transformation T,
which is also a transformation from Rn to Rm. I'm going to define right now
what it means to add the two transformations. So this is a definition. Let me write it as
a definition. I'm going to define the
addition of our two transformations. So if I add our two
transformations, the addition of two transformations operating
on some vector x, this is a definition. I'm going to say this is the
same thing as the first transformation operating on the
vector x plus the second transformation operating
on the vector x. And obviously, this is going
to end up being a vector in Rm, so this whole thing is going
to be a vector in Rm. By definition, this S plus T
transformation is still a transformation because it
takes an input from Rn. It's still a transformation
from Rn to Rm. Now let me make another
definition. Let me define -- I'll
do it in green. Maybe I'll do it in purple. I'm going to define a scalar
multiple of a transformation. So I'm going to define, let's
say c, where c is just any real number. c times the transformation S of
some vector x, I'm going to say that this is equal to c
times the transformation of x. And so similarly, the
transformation of x obviously is going to be in Rm. So if you multiply any vector
in Rm times some scalar, you're still going to have
another vector in Rm. So luckily for us, this
definition of a scalar multiple-- so if I have this
new transformation called c times S, this is still a
mapping from Rn to Rm. This is still a vector
in Rm and this is still a vector in Rn. Fair enough. Now, let's see what happens if
we look at their corresponding matrices for these
transformation. We've seen in a previous video
that any linear transformation can be represented as a
matrix vector product. So let's say that S of a vector
x is equivalent to the matrix A times that vector x. And let's say that T of x is
equal to the matrix B times the vector x. And, of course, since both of
these guys are mappings from Rn to Rm, both of these
matrices are going to be m by n. Both of these are
m by n matrices. Now, let's just go back to these
definitions that I just constructed. What is S of T of x? That can then be written as--
so let me write it this way. I'll do it in that same color. So you have S-- I was going
to do it in red. Maybe I'll do it right here. You have S plus T-- that's
a capital T. S plus T of x-- I'm just
re-writing this up here -- is equal to S of x plus T of x, or
the transformation T of x, which we now know is equal
to these two things. So this is equal to this
term right there. The transformation S of
x is equal to Ax. That's that one right there. And then the transformation
T of x is equal to B, the matrix B times x. Now, what are these things? Let me write our two matrices in
a form that you're probably familiar with right now. Let's say the matrix A is just a
bunch of column vectors: a1, a2, all the way to an. And similarly, the matrix
B is just a bunch of column vectors. The matrix B is b1, b2,
all the way to bn. These are each column vectors
with m components, one for each of the rows, and there's n
of these because there are n columns in each of
these vectors. So when you multiply this
guy times-- let me make it very clear. If I multiply an x, the vector
x is going to look like this. The vector x is going
to be x1, x2, all the way down to xn. And we've shown this multiple,
multiple times. It's a very handy way
of thinking about matrix vector products. But we know that this product
right here can be written be as each of these scalar terms
in x times its corresponding column vector in A. I've done this, and it's
probably the fifth video that I'm doing this. So this can be written as x1, x1
times a1 plus x2 times a2, all the way to xn times
an is equal to this. That's what ax can be rewritten
as, as kind of a weighted combination of these
column vectors where the weights are each of the values
of our vector x. And I have to add
this guy to bx. So bx, by the same argument, so
plus is just going to be-- let me do it in the blue. It's going to be x1 times b1
plus x2 times b2, all the way to xn times bn. Now, what is this equal to? Well, we know that scalar
multiplication times vector exhibits the distributive
property, so we can just add these two guys right here
and factor out the x1. And what do we get? We get this is equal to-- this
whole expression right here, let me draw a line here, because
I'm not saying this matrix is equal to that. I'm saying that this is equal to
this, is equal to this term plus this term, which is equal
to x1 times a1 plus b1, plus x2 times a2-- I'm just adding
these two terms up-- x2 times a2 plus b2, all the way to
plus xn times an plus bn. So what is this thing
equal to? Well, this is equal to some new
matrix, and let's define this new matrix. This is equal to some new
matrix-- I'll make it pretty big right here-- times
our vector x . I'll do the vector x in green. Vector x we know is x1, x2,
all the way down to xn. But what is the new matrix
going to be? Well, this product is going to
be each of these scalar terms times the column vectors
of this matrix. So these guys right here are
the columns of my matrix. This thing is equivalent to a
matrix where the first column right here is a1 plus b1. We're essentially adding
the column vectors of those two guys. The second column right here--
let me draw a little line right there to show you that
these are different expressions. The second one would be a2 plus
b2, and then we'll just have a bunch of them, and
then the last one will just be an plus bn. So what happens is that, by
definition, when I added these two transformations,
I just used their corresponding matrices. And I said you know what? The addition of these two
transformations created a new transformation that is
essentially some matrix times my vector, and that matrix ended
up being the sum of the corresponding column vectors
of our two original transformation matrices,
right? This new matrix that I got, and
I haven't defined matrix addition yet, but we got
here just by thinking about vector addition. This matrix is constructed by
adding the corresponding vectors of the matrices
A and B. Now, why did I go through
all of this trouble? Well, I can make a new
definition here that'll make everything fit together well. I'm going to define this matrix
right here as A plus B. So my new matrix definition,
if I have two matrices that have the same dimensions, and
they have to have the same dimensions, I'm defining A plus
B to be equal to some new matrix where you add up their
corresponding columns. So a1 plus b1, just like what
I did here, I don't have to rewrite it, all the way up to an
plus bn is the last column. And you've seen this before in
your algebra II class, but I wanted here to do it, because
this shows you the motivation for it. Because now we can say
that the sum of two transformations, So S plus T of
x, which is equal to S of x-- this is a vector-- S of x
plus T of x, which we know is equal to A times x plus B times
x, we can now say is equal to, because it's equal to
some new matrix, which we can now call A plus
B times x, right? I just showed this part is from
the definition of our transformations into some of
our transformation that I defined earlier in this video. And then when we just worked
this out and kind of expressed these products as products of or
as weighted combinations of the column vectors
of these guys, we got to this new matrix. And I defined this new
matrix as A plus B. And I did that because it has
this neat property now because now the sum of two linear
transformations operating on x is equivalent to, when you think
of it is a matrix vector product, as the sum of
their two matrices. Now, let's do the same thing
with scalar multiplication. We know that c times our
transformation of x by definition I'm saying is c times
the transformation of x. So c times whatever vector
this is in Rm. And so we know that S of x can
be rewritten as Ax, so this is c times A times x. And we know that Ax can be
rewritten as this is equal to c times x1 times the first
column vector in a, so a1 plus x2 times a2,xn all the way
to plus xn times an. Now, what is this? This is just scalar
multiplication. We can just distribute this c. and then what do we get? We get x, and multiplication
is associative. c is a scalar, x1 is a scalar,
so we can switch them around if we want. We know that scalar
multiplication is distributive, so we can write
this as x1 times ca1 plus x2 times ca2, all the way
to xn times can. Now, what is this equal to? This is equal to some
new matrix times x. This is equal to some new
matrix-- let me make that here-- times x1, x2,
all the way to xn. And what is that new matrix? What are the columns
of the new matrix? Well, the columns are
now that, that, all the way to that. So the columns of this new
matrix are ca1, ca2, all the way to can. Now, why would I go through
this exercise? Well, wouldn't it be nice,
I already said that by definition a scalar multiple of
a transformation is equal to the scalar times a
transformation of any vector that you input into it. And, of course, that is
equal to c times Ax. Now, wouldn't it be nice if I
could define this thing as some new matrix times
a vector x, right? Because this should also be
a linear transformation. And this new matrix I'm
going to define. This is a definition again. I'm going to define this new
matrix as being c times A. So now we have this definition
that c times A, if I take any scalar times any matrix A, it's
just equal to c times each of the column vectors. And we know what happens when
you take a scalar times each of the-- just let
me write this. This is equal to c times a1, c
times a2-- I'm just rewriting what I just wrote there-- all
the way to c times an. But what is this in effect? We know that when you multiply c
times a vector, you multiply the scalar times each of
the vector's elements. So this is the equivalent of
multiplying c times every entry up in this matrix
right here. And with this video, you know,
you're probably saying, hey, Sal, I already knew how to-- in
algebra II in tenth grade or ninth grade, I already was
exposed to multiplying a scalar times a matrix or adding
two matrices with the same dimensions. Why did you go through all of
this trouble of the defining the sum of transformations
and the sum of matrices? And I went through the trouble
because I wanted you to understand that there's
nothing-- I mean, it is natural, but there's nothing
about the universe that said matrices had to be
defined this way. Matrix addition, or matrix
scalar multiplication, or the addition of two transformations. I wanted you to see the
mathematical world has constructed it in this way
because it seems to have nice properties that are useful. And that's what I've
done in this video. In the next video, I'll
do a couple of scalar multiplications and matrix
additions just to make sure that you remember what you had
learned in your ninth or tenth grade algebra class, but you'll
find that the actual operations are almost
trivially simple.