So let's continue looking at linear transformations and how to represent linear transformations
as a matrix. In this particular example, we're going look at a linear transformation. That's
what LT stands for. We're going to note this linear transformation by T. And this linear
transformation takes vectors in R3 and outputs a vector in R4. So the inputs to this linear
transformation T or all of length 3, but the outputs are all of length 4. And we're told
some interesting information about this linear transformation.
We're told the following. We're told that the linear transformation of the vector e1
results in this four element vector, 3, 2, negative 1, and 0. When we use e1, that's
actually a very special vector. E1 is always a vector whose first element, that's why it's
subscript 1, e sub 1, whose first element is 1 and every other element of the vector
is 0. Since this linear transformation takes elements in R3, then we know that e1 is going
to be a vector of length 3. So up here, we've written down e1, it's going to be of length
3. And its first element is going to be 1 and all other elements are going to be equal
to 0. In addition to this, we're also told some
other things about the transform involving e2 and e3. So let's go ahead and write those
down. E2 is defined similarly, since we're working in R3, it's going to be a length 3
vector, whose second element is 1, and the same thing for e3. It's going to be a length
3 vector, because we're working on R3. And it's third element will be 1 and all other
elements will be 0. That's the definition of e3. So we're also told that the transformation
of e2 is equal to this vector. And the transformation of e3 is equal to this vector. So we know
some information about this linear transformation. The first question we're asked is to find
the standard matrix representation of this linear transformation. Anytime we're dealing
with linear transformations, we can find a matrix representation of these linear transformation.
By definition, the standard matrix representation is always equal to a matrix of the following
form-- the first column of the matrix representation is the linear transformation operating on
e1. The second column is the linear transformation operating on e2. The third column is the linear
transformation operating on e3. And we would keep going in general.
The fourth column is the linear transformation operating on e4, fifth column on e5, et cetera.
For this particular example, though, if we apply the definition of the standard matrix
representation, we only have to do three columns, because this is a linear transformation that
takes things of length 3 and outputs things of length 4. So we're only going to have three
columns here. So these individual linearly transform vectors are actually provided in
the problem statement. So constructing A is almost trivial. It's really just applying
the definition of what we mean by the standard matrix representation.
So I can just write down the first column is this vector right here, the second column
is this vector right here, and the third column is this vector right here. So this is the
linear-- I'm sorry, the standard matrix representation of this linear transformation. We can check,
just to make sure we did this right. It's almost trivial to check, but we can check.
What does it mean to take this linear transformation of e1? Well, in terms of the matrix formulation,
it means take A times e1. If I take A times e1, I write down A. I multiply it by e1. E1
is 1, 0, 0. We know how to do this matrix multiplication.
This has to take 1 times the first column plus 0 times the second column plus 0 times
the third column. So essentially, this matrix vector product is going to return just the
first column, which is exactly what we were told up here that the linear transformation
of e1 was equal to. So it's almost trivial to check, but we checked it anyway just to
make sure everything is self in system. So everything makes sense here.
Let's do something a little bit more interesting. We know how to write our linear transformation
in terms of a matrix representation. Let's deal with this vector now. Consider x equals
2 negative 1, 3. So here's a length 3 vector. We would like to compute the linear transformation
of this vector. So how do we go about doing this? I could just plug-in and compute Ax,
that would be one way to do it. And that'd be perfectly fine. The way I'm going to go
ahead and do it and this problem is to exploit the fact that we are dealing with a linear
transformation. So to use that fact, first note that we can
write e-- I'm sorry, we can write the vector x in this way. We can write it as 2 times
e1 minus e2 plus 3 times e3. So we're just writing the vector x in terms of these convenient
bases the letters represent x in terms of these other vectors e1, e2, and e3. Because
remember you want only has a one in the first element, so that's why we chose this 2 right
here. E2 only has a 1 in the second element, so we can get the negative 1 here. And e3
has a 1 only in the third element, so 3 times that gives us that 3 right there.
So let's go ahead and compute T of x. Well, to compute T of x, equivalently, I need to
compute the linear transformation of this quantity, all right, as x is actually equal
to that quantity right there. And now, here's what's interesting. This is equal to, because
it's a linear transformation. You can't do this every time, but because T is a linear
transformation, linear transformations have certain properties. They have a property that
the transformation of a sum is equal to the sum of the transformations. And also, any
scale factors can be pulled outside the linear transformation.
So this turns into a sum of 2 times the transform of e1 plus a negative 1 times a transform
of e2 plus 3 times the transformation of e3, because I'm dealing with a linear transformation.
There's really just the definition of what it means to be linear. This is something we
know. This is something we know. And this is something we know. So let me go ahead and
plug those in. 2 times the transformation of e1 is this minus the transformation of
e2 plus 3 times the transformation of e3. We know how to do scalar multiplication. I
just multiply everything by 2, so I get a 6, 4, negative 2, 0 minus this vector plus
3 times this vector is 3, 0, 0, 6. Now if I add all these up, 6 minus 9 plus
3 gives me 0, 4 minus 4 gives me 0, and negative 2 minus 5 is a negative 7, and 0 plus 0 plus
a negative 6 is a negative 6. So here is the linear transformation of the vector x. I could
have computed it just by computing A times x. That'd be one way to do it, but we went
ahead and did it using the definition of what it means to be a linear transformation. This
means that superposition and homogeneity hold and we can write this equals this. The transformation
of a sum is as a sum. Of transformations and scale factors can be pulled outside of the
linear transformation operation.