Let's go ahead and take a look at one more example of finding a matrix representation
of a linear transformation.
In this particular problem, we're going to phrase the problem statement just a little
bit different again.
What we're going to assume is that we've got some matrix that has all these entries that
we don't know what they are.
And we know that when operating on the vector x, which has three elements, x1, x2, and x3,
that gives us the following vector.
The vector has a first element of 3x1 minus x3.
The second element of the vector is minus 2x1 plus 4x2.
And then the final element of the vector is 6x2 minus x3.
And what we don't know, like I said, is what are these entries here of the A matrix.
So A is a 3 by 3 matrix.
And we don't know what those entries are.
That's where we'd like to find and we know that it has to equal this when operating on
the vector x.
And this has to hold for all possible values of the vector x.
So that's what we mean here hold for all, that's the for all, symbol for all, x1, x2,
and x3.
So for an arbitrary vector x.
So what I'm going to think about this is I'm going to think about this as the vector V1,
so I'm going to think of this first column as V1, I'm going to think of the second column
as V2, and the third column as V3.
When written that way, we know how to do matrix vector multiplication.
Each element of the vector x weights corresponding column of the matrix A, so I can actually
write out what this product is equal to.
Its x1 times the vector V1 plus x2 times the vector V2 plus x3 times the vector V3.
And we need that to equal the quantity that we were given.
Let's think about this.
Let's replace each of these vectors V1, and V2, and V3 with the empty vectors that we
don't know what they are, right?
So V1 is this unknown quantity, so I've just left it blank right here, same thing for V2
and V3.
But now, by comparing elements over here to the corresponding entries and variables over
here, we can figure out what needs to go there.
For instance, on this first vector, when I look up here at the first element, what contributes
to this first element on the right are all the first elements of each vector on the left.
So when I look over here on the right, I see that I have 3x1's.
The only way to get x1's are from this vector and specifically, this first element of it.
So I have to have a 3 there to end up with 3x1 in the first element of the final vector.
Similarly, when I look right here, I don't see any x2's, so I'm going to end up having
a 0 here.
Let's go ahead and keep filling it out for x1 right now.
If we go to the second element, how many x1's do I have here?
Well, I have a negative 2.
That's why I put a negative 2 right here.
And then when I looked down here at the bottom, how many x1's do I have there?
Well, I don't have any, so I've got 0.
So we can just fill out this vector by inspection, because we know how vector addition is going
to work.
We add all three of these vectors up and we know what it needs to equal to.
So if we go ahead and do the same thing for the x2 vector that-- I'm sorry, the vector
that's multiplying the scalar x2 is the V2, when I look at the first entry, I don't have
any x2's.
It says no x2's right here.
Right here, I have 4x2's, so 4 needs to go there, and then a 6 needs to go here.
And then finally for x3, I have negative 1, 0x3 and minus 3 x3.
So we've been able to fill this out and figure out what each element of these vectors needs
to be.
So if I put this together now, now that I know V1, V2, and V3, we know that A has a
first column of V1, has a second column of V2, and a third column of V3.
So that is the matrix representation of this linear transformation.