Matrices and linear transformations For grade 1, undergraduate students For grade 1, undergraduate students Made by Department of Math.,Anqing Teachers college

1. Representing a linear transformation by a matrix L et us suppose given a linear transformation between the finite-dimensional vector spaces V and W. Let be an ordered basis for V and an ordered basis for W. It is then possible to fine unique numbers

such that which we have seen may be written more compactly as

The matrix is called the matrix of T relative to the ordered bases EXAMPLE 1. Let be the linear transformation given by Calculate the matrix of T relative to the standard basis of.

Solu t ion. By the “standard basis of ”, we of course mean the basis and that we are to use this ordered basis in both the domain and range of T. We have so the matrix we seek is

EXAMPLE2. With T as in Example 1, find the matrix of T relative to the pair of ordered bases and, where Solution. We still have but now we must write these equation as

so that is the matrix that we now seek. EXAMPLE3. Calculate the matrix of the differentiation operator D: relative to the usual basis for.

Solution. We have for and Thus the matrix that we seek is

Remark. There is of course no reason to restrict us to square matrices. For example we have the linear transformation and we can ask for its matrix relative to the standard bases of and. Question. What size is this matrix? How to calculate the matrix of linear transformation

2. An isomorphism and its matrix Proposition. is an isomorphism iff its matrix A is invertible. PROOF. Suppose that T is an isomorphism Let be the linear transformation inverse to T. Let B be the matrix of S relative to the basis pair (Note that we have interchanged the role of the bases ).

Thus if B =( ) then. Since the matrix product AB is the matrix of the linear transformation relative to the basis pair But for all since T and S are inverse isomorphisms. In particular

and hence the matrix of relative to the bases is Therefore AB = I. Likewise, since the matrix product BA is the matrix of the linear transformation

relative to the basis pair- But for all in V because S and T are inverse isomorphisms, and hence as before we find BA=I. This shows that if is an isomorphism then a matrix S for T is always invertible.

To prove the converse, we suppose that the matrix A of T is invertible. Let B be a matrix such that AB=I=BA. Let be the linear transformation whose matrix relative to the ordered bases is B. Then the matrix of T relative to the ordered bases is

Therefore and I have the same matrix relative to the bases so that by, that is for all in V.

Likewise we see that for all in V, so that S and T are inverse isomorphisms. EXAMPLE 4. Find the matrix of the identity linear transformation relative to the ordered bases

Solution. We have So the matrix we seek is

Remark. In view of Example above, it is reasonable to expect that when we calculate with matrices of transformations, we insist upon using the same ordered basis twice to do the calculation, rather than work with distinct ordered bases

EXAMPLE 5. Let be the linear transformation given by Calculate the matrix of T relative to (a) the standard basis of (b) the basis used twice.

Solution. To do Part (a) we compute as follows: Thus the desired matrix is To so the computations of Part (b), let us set

Then we find So the matrix for Part (b) is

3. Matrices relative to different bases Theorem. Let A and B be matrices, V and n -dimensional vector space and W an m -dimensional vector space. Then A and B represent the same linear transformation relative to (perhaps) different pairs of ordered bases iff there exist nonsingular matrices P and Q such that where P is and Q is.

PROOF. There are two things we must prove. First, if A and B represent the same linear transformation relative to different bases of V and B represent the same linear transformation relative to different bases of V and W we must construct invertible matrices P and Q such that we must construct a linear transformation and pairs of ordered bases for V and W such that A represents T relative to one pair and S relative to the other. Consider the first of these. We suppose given bases

such that the matrix of T relative to these bases is A, and bases, such that the matrix of T relative to these bases is B. Let P be the matrix of relative to the bases

Then by the proposition above, P is invertible., relative to the bases Then is also invertible and represent the matrix of Let Q be the matrix of relative to he bases

Therefore PB is the matrix of relative to the bases If we apply it again we see that is the matrix of T relative to the bases

But Q is also the matrix of T relative to the bases To prove the converse, suppose given invertible matrices P and Q such that so that as required.

Choose bases for V and W respectively. Let be the linear transformation whose matrix is A relative to these bases. Let, since P and A are isomorphisms, the collections

are bases for V and W respectively. A brute force computation now shows that B is the matrix of T relative to the bases EXAMPLE 6. Recall that we are given the linear transformation defined by

and is the matrix of T relative to the standard basis of, while is the matrix of relative to the ordered basis

of Since there are invertible matrices P, and such that, our task is to

compute P and. We compute them as follows. (1) P is the matrix of relative to the basis pair and (2) is the matrix of relative to the basis pair and

The computation of P is easy and gives us The computation of is not hard and depends on the following equations

so that A tedious computation shows that.

That is

4. Some exercises 1. Find the matrix of following linear transformations relative to the stand are bases for (a) given by (b) given by

(c) given by (d) given by

2. Let be the linear transformation whose matrix relative to the standard bases is Find

3. Let be the linear transformations with matrices respectively. What is the matrix of the linear transformation Find (3 S-7T )(1,2,3).

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