233 Problem Sets

Problem Set 1

1.2 1, 3, 5
1.3 3, 5
1.4 7, 9
2.1 5, 7

Now that you are done, here are solutions to problem set one.

Problem Set 2

2.2 7, 13
2.3 9, 11
2.4 3, 7
2.5 5, 7
2.6 3, 7

You do what you do; here are solutions to problem set two.

Problem Set 3

3.1 5, 11
3.2 3, 7
no problems from 3.3
3.4 7, 11
3.5 5, 9

Look and you will see, solutions to problem set three.

Problem Set 4

4.1 6, 9
4.2 5, 8
4.3 6, 10
no problems from 4.4 nor 4.5

Just wait, there's more, solutions to problem set four.

Problem Set 6

6.1 5, 6
6.2 6, 7
6.3 7, 8
6.4 1, 4

We skipped five in this mix, so now are solutions to problem set six.

Problem Set VS

The problems will come from here.

Choose one of 1-4.

5-6 as one (note a = {a_n} not {a+n} as written in the poorly edited book)

26-27 as one

44 (there are four vectors, not three. Did I mention was poorly edited?)

62-67: find one that is a subspace, and find a basis for that subspace. find one that is not and explain why not. This counts as two problems.

As a setup - from when you were in HS, what were linear polynomials? Now, the real topic: consider some functions T: P -> P (these are functions that take a polynomial to another polynomial): 1. take polynomial p and send it to p(a) for a real number a. 2. take p and multiply by a polynomial q. 3. take p and add 1. Show all work to explain which ones are linear transformations from P to P and which are not. Are there any surprises compared to your original answer from HS?

Here is a transformation from M³ -> M³ (three by three matrices). Take matrix A and send it to A^T + A. Show that this is a linear transformation. What is a basis for the kernel (= "nullspace" or "null space") of this transformation? What is a basis for the image (= "range" or "column space")? What are the dimensions of each? With respect to Erin's basis (the standard basis) for M³, what is the matrix of associated to this transformation with this basis for both domain and target (="codomain").

By my count this is 8 problems, please be sure that you see that.

Now that we've made it through all of our paces, here are the solutions for vector spaces.