3. (a) Suppose that T: VV is a linear transformation, where V is a vector space of dimension n. If B = {₁,..., Un} is a basis of V, explain how to write the matrix of T with respect to B. (b) Give an example, with explanation, of a matrix in M2 (R) that is not diagonalizable. (c) What does it mean to say that two square matrices are similar? Determine, with explanation whether the following two matrices in M2 (R) are similar. [14] (d) Let A = -1 1 1 4 0-2 . Find a 3 x 3 invertible matrix PE M, (R) for which 4-3 -1 1 0 0 0 02 00-5 (Note: you can do this without calculating the characteristic polynomial of A.) P-¹AP =
3. (a) Suppose that T: VV is a linear transformation, where V is a vector space of dimension n. If B = {₁,..., Un} is a basis of V, explain how to write the matrix of T with respect to B. (b) Give an example, with explanation, of a matrix in M2 (R) that is not diagonalizable. (c) What does it mean to say that two square matrices are similar? Determine, with explanation whether the following two matrices in M2 (R) are similar. [14] (d) Let A = -1 1 1 4 0-2 . Find a 3 x 3 invertible matrix PE M, (R) for which 4-3 -1 1 0 0 0 02 00-5 (Note: you can do this without calculating the characteristic polynomial of A.) P-¹AP =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. (a) Suppose that T: VV is a linear transformation, where V is a vector space of dimension n. If
B = {₁,..., Un} is a basis of V, explain how to write the matrix of T with respect to B.
(b) Give an example, with explanation, of a matrix in M2 (R) that is not diagonalizable,
(c) What does it mean to say that two square matrices are similar? Determine, with explanation whether
the following two matrices in M₂ (R) are similar.
(d) Let A =
4 4
-1
1 1
4 0-2 . Find a 3 x 3 invertible matrix PE M₁ (R) for which
4-3 -1
1 0 0
02
00-5
0
(Note: you can do this without calculating the characteristic polynomial of A.)
P-¹AP =
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