In this video, we're going to work with a linear transformation.
And since it's a linear transformation, that means that we can find a matrix representation
of the linear transformation.
So the linear transformation we're going to work with here, we're going to denote T. And
this is a linear transformation-- that's what LT stands for-- that takes us from R3 to R3.
So the inputs to this linear transformation are vectors in the reals of length 3.
And the output of the linear transformation is also a vector in the reals of length 3.
And we're told that we have this vector whose length is 3, whose elements are x1, x2, and
x3.
And we are also told that we are dealing with the vectors of v1, v2, and v3.
So these are the vectors that we're dealing with.
And we're going to do is, in part a, we are told that the linear transformation T of x
can be written like this.
Any vector x that you give it-- x is very arbitrary here, elements x1, x2, and x3.
The output of the linear transformation is simply the summation of the vectors v1, v2,
and v3, where each one has been weighted by the corresponding entry in x.
So the linear transformation of the vector x is x1, which is a scalar times the vector
v1, plus x2, which is a scalar times the vector v2, plus x3, which is a scalar times the vector
v3.
So if this is how this linear transformation operates, what is a?
How do we represent this linear transformation as a matrix operation on a vector x?
So we would like to be able to write our linear transformation like this.
We would like to be able to write T of x is equal to some matrix A, which we don't know,
times the vector x.
That's what we would like to do.
We would like to find out what this matrix A is.
Well, look at this.
If A happens to be the matrix comprised of the vectors v1, v2, and v3 as the columns--
so v1 is the first column, v2 is the second column, and v3 is the third column-- then
we can go ahead and compute what this would be.
So A x would be v1, v2, to v3 times the vector x.
Well, we know how to do matrix vector multiplication.
Each entry of the vector x multiplies each column vector of the matrix A. So this is
equal to v1x1, the first element of x times the first column of A, plus x2 times the second
column of a plus x3 times the third column of A.
So I wrote this a little different.
It probably would've been more appropriate to write it as x1v1, and x2v2, and x3v3, because
that's how we define scalar multiplication.
But it's clear here what we're doing.
This is just the vector v1 with every entry scaled by x1, et cetera.
So this is equal to x1v1, plus x2v2, plus x3v3. this is exactly what we wanted to achieve.
We wanted to have x1v1, x2v2, and x3v3.
So I kind of took a stab at what I thought A1 might be, and it turns out this matrix
right here is exactly what it needs to be.
We constructed the matrix A, and it turns out it yielded the exact transformation that
we wanted.
So A should be the matrix whose first column is v1, whose second column is v2, and whose
third column is v3.
This is the matrix representation for the linear transformation defined by this operation
right here.
Let's do another little example.
What if T of x, the linear transformation, was defined like this?
This a little bit different.
Now, I'm saying that the linear transformation takes an input vector x and it returns the
x1 times v1 plus x2 times v2.
This is a little bit different.
There is no x3 or v3 here.
It's only x1v1 and x2v2.
So, in this case, what would A be?
Well, it's pretty easy to guess what it should be.
Let's take a stab at it and say that A is the matrix whose first column is v1, whose
second column is v2, and whose third column is the all zero vector.
So this is a vector of zero, the column vector.
And for this particular example, it has three zeros in it.
If I multiply this by my vector x, what do I get?
I get x1v1 plus x2v2 plus x3 times the all zero vector, which gives us an all zero vector,
which when added to these other vectors doesn't change anything.
It just goes to zero.
So I end up with just x1v1 plus x2v2.
So I've kind of guessed what I think A should be, and we have verified that it does give
us the transformation of x as stated in the problem right here.
So for this example, A would be 1, 2, 3, 0, negative 1, 4-- because that's what v2 is--
and then it's third column would be the all zero vector.
So this was a little simple example-- actually two examples-- of when you're given the equation
for how the linear transformation works, how you go about kind of guessing at the form
of A. Often A is comprised of the vectors that you're given.
You just kind of have to multiply it out and verify that that is the case.
For more complicated examples, you might have to actually kind of write things out and solve
the system of equations.
But for this particular example that we looked at, it was fairly obvious how to write down
a guess for A and then verify it doing very simple matrix vector multiplications.