Math 2586 sample Final

Hsian-Hua Tseng

December 2017

Show your work!

Problems

1. Give the detailed deﬁnition of vector spaces.

2. Consider the following 3 ×3 matrix

A=

111

121

112

.

Find an invertible 3 ×3 matrix Sand a diagonal 3 ×3 matrix Dsuch that

AS =SD.

3. Let Abe a 3 ×3 matrix with 3 distinct eigenvalues. Denote by

c0+c1x+c2x2+c3x3

the characteristic polynomial of A. Show that

c0I3+c1A+c2A2+c3A3=O.

4. Determine whether the following matrix is invertible or not:

A=

111

121

112

.

If Ais invertible, compute A−1.

5. Show that the set of polynomials in xof degree at most 5 is a vector space.

6. Give the deﬁnition of orthogonal matrices.

1

## Document Summary

Problems: give the detailed de nition of vector spaces, consider the following 3 3 matrix. Find an invertible 3 3 matrix s and a diagonal 3 3 matrix d such that. As = sd: let a be a 3 3 matrix with 3 distinct eigenvalues. Denote by c0 + c1x + c2x2 + c3x3 the characteristic polynomial of a. Show that c0i3 + c1a + c2a2 + c3a3 = o: determine whether the following matrix is invertible or not: If a is invertible, compute a 1: show that the set of polynomials in x of degree at most 5 is a vector space, give the de nition of orthogonal matrices.