I am "scared" (to put it mildly) of these problems, which I need to review before my final tomorrow. Just to let all of you know, this is not homework. There are 25 or so problems, and I only understand around 10 of them.(adsbygoogle = window.adsbygoogle || []).push({});

Help me! I need an A on the final to get a B in the class.

1. If I: W-->W is the identity linear operator on W defined by I(w) =wforwin W, prove that the matrix of I repect with to any ordered basis T for W is a nXn I matrix, where dim W= n

2. Let L: W-->W be a linear operator defined by L(w) = bw, where b is a constant. Prove that the representation of L with respect to any ordered basis for W is a scalar matrix.

3. Let X,Y, Z be sqaure matrices. Show that: (a) X is similar to Y. (b) If X is similar to Y then Y is similar to X. (c) If X is similar to Y and Y is similar to Z, then X is similar to Z.

4. Show that if X and Y are similar matrices then X^k and Y^k are similar for any positive integer k.

5. Show that if X and Y are similar, then transpose(X) and transpose(Y) are similar.

6. X and Y are similar. Prove: 1) If X is nonsingular, then Y is nonsingular. 2) If X is nonsingular, then A^-1 and B^-1 are similar.

7. Prove that A and transpose(A) have equivalent eigenvectors. How are the associated eigenvectors of A and transpose(A) related?

8. Prove that if A^k = O for some positive k (A is a nilpotent matrix then, right?) then 0 is the only eigenvalue of A.

In 9, X is a mxm matrix

9a. Show that det(A) is the product of the roots of the characteristic polynomial of A.

9b. Show that A is singular if and only if 0 is an eigenvalue of A.

9c. Prove that if L: V--->V is a linear transformation, show that L is not one-to-one if and only if 0 is an eigenvector of L.

10. Show that if A and B are orthogonal matrices, then AB is a orthogonal matrix.

11. Show that if A is an orthogonal matrix, then det(A) is +/- 1.

Please everyone, I know that you may be annoyed at me for posting so many problems, but my final is tomorrow, and I am getting mental blocks on these problems. So, please, PF Members, will you also show your work in detail, so I can follow it? There's no solution pack for my review WS.

Again, detailed proofs would be really appreciated. I am so stressed about the final. :surprised

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# Some linear algebra proofs I couldn't figure out: Help!

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