Page 649 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

for all x in V, then .

Let l be a line in the -plane that passes through the origin and makes an angle θ with the positive x-axis. As illustrated in

18. the accompanying figure, let be the orthogonal projection of onto l. Use the method of Example 3 to show

that

Note See Example 6 of Section 4.3.

Figure Ex-18

Indicate whether each statement is always true or sometimes false. Justify your answer by giving a
19. logical argument or a counterexample.

(a) A matrix cannot be similar to itself.

(b) If A is similar to B, and B is similar to C, then A is similar to C.

(c) If A and B are similar and B is singular, then A is singular.

(d) If A and B are invertible and similar, then and are similar.

Find two nonzero matrices that are not similar, and explain why they are not.
20.

Complete the proof by filling in the blanks with an appropriate justification.
21.

Hypothesis: A and B are similar matrices.

Conclusion: A and B have the same characteristic polynomial (and hence the same
eigenvalues).

Proof: (1) _________

(2) _________

(3) _________