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Matrices and linear transformations For grade 1, undergraduate students

Matrices and linear transformations For grade 1, undergraduate students. Made by Department of Math. ,Anqing Teachers college. L et us suppose given a linear transformation between the finite-dimensional vector spaces V and W.

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Matrices and linear transformations For grade 1, undergraduate students

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

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

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

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

  5. Solution.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

  6. 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

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

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

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

  10. 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 ). 2. An isomorphismand its matrix

  11. 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. Thus if B=( ) then . In particular

  12. 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

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

  14. 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 To prove the converse, we suppose that the matrix A of T is invertible. Let B be a matrix such that AB=I=BA.

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

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

  17. Solution. We have So the matrix we seek is

  18. 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

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

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

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

  22. 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 .

  23. 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

  24. 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

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

  26. 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

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

  28. 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 ,

  29. 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

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

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

  32. 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

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

  34. so that A tedious computation shows that .

  35. That is

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

  37. (c) given by (d) given by

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

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

  40. Thanks !!!

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