Reference no: EM131075815
MATH 54 QUIZ 7-
1. Find an invertible matrix P and a matrix C of the form
such that A =
has the form A = PCP-1.
2. Give an example of a 2 × 2 matrix (with real coefficients) that has no eigenvectors in R2.
3. Suppose x is an eigenvector of A corresponding to an eigenvalue λ.
(i) Show that x is an eigenvector of 5I - A. What is the corresponding eigenvalue?
(ii) Show that x is an eigenvector of 5I - 3A + A2. What is the corresponding eigenvalue?
4. Give an example of a vector in R4 orthogonal to v = [1, 4, 2, 6]T.
5. Give an example of three vectors u1, u2, u3 in R2 such that u1 · u2 = 0 and u2 · u3 = 0 but u1 · u3 ≠ 0.
6. Let
. Compute the distance from y to the line through u and the origin.
7. Does there exist an orthonormal matrix 
such that |a1| + |a2| + · · · + |a9| > 3? If yes, give an example; if no, give a proof explaining why.
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