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MATH 108B PRACTICE FINAL
发布时间:2023-12-16
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MATH 108B
PRACTICE FINAL
Note: You may use without proof any of the results from lectures or homeworks, but in that case you should clearly state what you’re using.
Problem 1. Consider the linear transformation
![](/Uploads/20231216/657d4a00a42b3.png)
and let A 2 M3x3 (R) be the standard matrix of T.
(a) Determine whether A is diagonalizable, and if so, ind an invertible matrix S such that S-1 AS is diagonal.
(b) Let B = [T]B be the matrix of T associated to an arbitrary basis B of R3 . Is B diagonalizable? Justify your answer.
Problem 2. Let T : V ! V be an invertible linear transformation.
(a) Show that if λ is an eigenvalue of T, then λ 0, and λ -1 is an eigenvalue of T-1 .
(b) Show that for any eigenvalue λ of T, we have the following equality of eigenspaces:
E (T) = E - 1 (T-1).
Problem 3. Find values of a,b, c 2 R such that
is an orthogonal matrix.
Problem 4. Let A 2 Mnxn (R) be a symmetric matrix, and consider Rn with the dot product. Show that if ~v1 , ~v2 2 Rn are eigenvectors of A with diferent eigenvalues, then ~v1 and ~v2 are orthogonal.
(Hint: Use the formula for the dot product in terms of matrix multiplication.)
Problem 5. Let J be the Jordan canonical form of
Find J and an invertible matrix S such that S-1 AS = J.