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.

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

- Without looking in text, define Null Space, Column Space, Range, and Kernel
- 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

- Without looking in text, define linear independence, basis, standard basis.
- Can a set of 6 or more vectors in
**R**^{5}be linearly independent? Why? Can a basis for**R**^{5}have 6 or more elements? Why? - Can a set of 4 or fewer vectors in
**R**^{5}be spanning set for**R**^{5}? Why? Can a basis for**R**^{5}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:

- Suppose matrix
*A*has an eigenvalue of 2 with the corresponding eigenvector**v**= [1 1 3]^{T}. Compute*A*^{3}**v**,*A*^{7}**v**, and*A*^{k}**v** - Suppose that the same matrix
*A*in part 1 has eigenvalue of 1/4 with corresponding eigenvector**w**= [2 0 1]^{T}. Compute*A*^{3}(**v + w**),*A*(^{7}**v + w**), and*A*(^{k}**v + w**) - Find a formula for
*A*^{k}**x**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

- 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**u**and**v**.

- Find a formula for
*A*[1 1]^{k}^{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*k*th and (*k*+1)st fibonacci numbers. - By direct calculation with
*k*= 6, verify that your formula produces the 6th and 7th fibonacci numbers.