PROFESSOR: Hi, everyone.
Today, we're going to talk about linear transformations.
So, we've seen linear transformations incognito all
the time until now.
We've played around with matrices.
Matrices multiplying vectors, say, in Rn and producing
vectors in Rm.
So really, the language of linear transformations only
provides a nicer framework when we want to analyze linear
operations on more abstract vector spaces, like the one we
have in this problem here.
We're going to work with the space of two by two matrices.
And we're going to analyze the operation, have the matrix A,
and we produce its transpose.
OK.
So please take a few minutes to try the problem on your own
and come back.
Hi, again.
OK.
So the first question we need to ask ourselves is, indeed,
why is T a linear operator?
So what are the abstract properties that a linear
operator satisfies?
Well, what happens when T acts on the sum of two
matrices, A and B?
So we've produced--
this is the matrix.
The transpose of A plus B. But we know that this is A
transpose, B transpose.
And so, this is exactly T of A plus T of B. So the
transformation that we're analyzing takes the sum of two
matrices into the sum of their transformations.
OK.
Similarly, it takes a multiple of a transformation into the
multiple of the transformations.
So it takes the matrix C times A to C times A transpose,
which
is C T of A. OK.
So it is a linear operator.
Now, can we figure out what its inverse is?
Well, what does the transpose do?
It takes a column and flips it into a row.
So what happens if we apply the operation once again?
Well, it's going to take the row and turn it back down to
the column.
So applying the transformation twice, we come back to the
original situation.
So therefore, t squared is the identity.
And from this, we infer that the inverse is the
transformation itself.
Now, this was part one.
Part two, we'll compute the matrix of the linear
transformation in the following two bases.
So the first basis is, in fact--
it is the standard basis for the space
of two by two matrices.
And the way we compute the matrix, we first compute what
T does to each of the basis elements.
So T of v1.
Let's go back.
So here.
So T takes the transpose of this matrix.
And we see that the transpose of 1, 0, 0, 0 is 1, 0, 0, 0.
So it's a symmetric matrix.
So T of v1.
Here's v1.
What about T of v2?
Come back here.
So this one comes here.
0 comes here.
And so we actually get v3.
So T of v2 is v3.
Similarly, T of v3 is v2.
And finally, T of v4.
Well, v4 is a symmetric matrix as well.
So the transpose doesn't change it.
OK.
Now, we encode this into a matrix in the following way.
Essentially, the first column will tell us how T of v1 is
expressed as a linear combination
of the basis elements.
Or in this case, it's just v1.
So it's going to be 1 times v1 plus 0, v2 plus
0, v3 plus 0, v4.
T of v2 is v3.
So we have 0, 0, 1, 0.
T of v3 is 0 v1, 1 v2, 0 v3, 0 v4.
And T of v4 is 0 v1, 0 v2, 0 v3, plus 1 v4.
OK.
So we've written down the matrix of the linear
transformation, T, in the standard basis.
And you can check that this is exactly what we want.
The representation of some matrix, say, 1, 2, 3, 4, in
this standard basis is, it's the vector 1, 2, 3, 4.
T takes this to its transpose 1, 3, 2, 4.
So this in the basis is represented as 1, 3, 2, 4.
Right?
And it's not hard to see that MT multiplies this vector, we
get exactly this vector.
So we'll pause for a bit, so that I erase the board.
And we're going to return with the representation of T in the
basis, w1, w2, w3, and w4.
OK.
So let's compute now the matrix T in the basis w1, w2,
w3, and w4.
We played the same game.
We look at how T acts on each of the basis vectors.
So T of w1--
well, w1 is a symmetric matrix.
So T of w1 is w1.
Similarly, with w2 and w3.
They're all symmetric.
What about w4?
Well, we see that the 1 comes down here.
The negative one comes up here.
I mean, in the end we just get the negative of the w4.
So, let me just write this out.
We had T of w1 equal to w1, T of w2 equal to w2, T of w3
equal to w3, and T of w4, was negative on w4.
So therefore, the matrix of the linear transformation, T,
in this basis.
I'm going to call the matrix M prime.
T has a fairly simple expression.
The only non-zero entries are on a diagonal.
And they're precisely 1, 1, 1, and negative 1.
And finally, let's tackle the eigenvalues slash
eigenvectors issue.
Well, you've seen what an eigenvector for a matrix is.
And the idea for an eigenvalue, eigenvector for a
linear transformation is virtually the same.
And we are looking for the vectors v and the scale is
lambda such that T of v is lambda of v. But if you guys
look back to what we just did with w1, w2, w3, and w4,
you'll see precisely that w1, w2, and w3 are eigenvectors
for T with eigenvalue 1.
And w4 is an eigenvector for T with eigenvalue negative 1.
So yeah, we essentially have solved the problem knowing a
very, very nice basis in which we computed the linear
transformation, T. So I'll leave it at that.