Linear Algebra Math 310
Assignment Sheet 2
This sheet specifies a selection of exercises from the text, and in
cases a few additional exercises. The notation 1-33o means the odd
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*,
Section 4.2 1-25o,27*,29*,30*,31*,33*,35*, plus:
- Without looking in text, define Null Space, Column Space, Range,
- Makeup a definition of the Row Space of a matrix in analogy with
definition of column space. Show that every for every v in
space of A and for every w in the null space of A,
dot product of v and w is 0.
Section 4.3 1-13o,14,15,19,(21-26,29-32)*,plus
- Without looking in text, define linear independence, basis,
- Can a set of 6 or more vectors in R5 be
Why? Can a basis for R5 have 6 or more elements?
- Can a set of 4 or fewer vectors in R5 be
set for R5? Why? Can a basis for R5
4 or fewer elements? Why?
Section 4.5 1-17o, 19ab, (19cde, 21, 23,
Section 4.6 1-25o,27-30
Difference Equations Handout
Section 5.1 Regular: 1, 3, 5, 7, 13, 15, 16, 17, 19,
Problems: 23*, 25*, 26* (In problem 26, an example of the sort of
consideration is given by A =
The following additional problems are regular:
- Suppose matrix A has an eigenvalue of 2 with the
eigenvector v = [1 1 3]T.
Compute A3 v, A7
and Ak v
- Suppose that the same matrix A in part 1 has eigenvalue
with corresponding eigenvector w = [2 0 1]T.
Compute A3 (v + w), A7 (v
+ w), and Ak (v + w)
- 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,
Optional Problems: 19*,
The following additional problems are regular. For all of these
let the matrix A be given by
- Find the eigenvalues of A
- For each eigenvalue, find one eigenvector. Call these
eigenvectors u and v.
- Express [1 1]T as a linear combination of
- Find a formula for Ak[1 1]T
the same methods as in the extra problems for section 5.1. (As
in this handout, the two
entries of this
vector are the kth and (k+1)st
- By direct calculation with k = 6, verify that your
the 6th and 7th fibonacci numbers.