Linear Algebra Math 310
Spring 2017
Assignment Sheet 2

This sheet specifies a selection of exercises from the text, and in some cases a few additional exercises. The notation 1-33o means the odd problems between 1 and 33, inclusive.

Section 4.1 1, 3, 9, 11, 13, 15, 17.  All the following are optional:  4*, 5*, 7*, 19*, 21*,  27*, (29-33)*

Section 4.2   1-25o,27*,29*,30*,31*,33*,35*, plus:

1. Without looking in text, define Null Space, Column Space, Range, and Kernel
2. Makeup a definition of the Row Space of a matrix in analogy with the definition of column space. Show that every for every v in the row space of A and for every w in the null space of A, the dot product of v and w is 0.

Section 4.3   1-13o,14,15,19,(21-26,29-32)*,plus

1. Without looking in text, define linear independence, basis, standard basis.
2. Can a set of 6 or more vectors in R5 be linearly independent? Why? Can a basis for R5 have 6 or more elements? Why?
3. Can a set of 4 or fewer vectors in R5 be spanning set for R5? Why? Can a basis for R5 have 4 or fewer elements? Why?

Section 4.5   1-17o, 19ab,  (19cde, 21, 23, 27,29-32)*

Section 4.6   1-25o,27-30

Difference Equations Handout  Problems 1-6

Section 5.1  Regular: 1, 3, 5, 7, 13, 15, 16, 17, 19, 21.  Optional Problems: 23*, 25*, 26* (In problem 26, an example of the sort of matrix under consideration is given by A = .)

The following additional problems are regular:

1. Suppose matrix A has an eigenvalue of 2 with the corresponding eigenvector v = [1  1  3]T.  Compute A3 v, A7 v, and Ak v
2. Suppose that the same matrix A in part 1 has eigenvalue of 1/4 with corresponding eigenvector w = [2 0 1]T.  Compute A3 (v + w), A7 (v + w), and Ak (v + w)
3. Find a formula for Akx where x = [4  2  7]T .  (Hint: express x as a linear combination of v and w).

Section 5.2   Regular: 1, 3, 7, 11, 15, 21.  Optional Problems: 19*, 21*.
The following additional problems are regular.  For all of these problems, let the matrix A be given by

A =
1. Find the eigenvalues of A
2. For each eigenvalue, find one eigenvector.  Call these eigenvectors u and v.
3. Express [1  1]T as a linear combination of u and v.
4. Find a formula for Ak[1  1]T using the same methods as in the extra problems for section 5.1.  (As discussed in this handout, the two entries of this vector are the kth and (k+1)st fibonacci numbers.
5. By direct calculation with k = 6, verify that your formula produces the 6th and 7th fibonacci numbers.