HMC Math 63: Linear Algebra II (Summer, 2001)
Homework

### Assignment #13 (due Never):

This is not a real assignment... just a few computational
problems that you could do to practice for the final.
NOTE: Some of these have appeared on previous assignments.

Section 4.2: 31
Section 4.4: 13
Section 4.7: 5
Section 5.4: 5, 11
Section 5.5: 5
Section 7.1: 17

### Assignment #12 (due Fri. 6/1):

Section 5.5: 3, 7, 15
Section 7.1: 13, 19, 23, 26, 31

### Assignment #11 (due Thu. 5/31):

Section 5.4: 11, 17, 23, 28, 31
Read as much of Section 7.2 as is necessary to do...
Section 7.2: 1, 3a, 5a, 27
11A: Let L:V->W be a linear transformation, and let dim V = dim W.
Prove that L is 1-1 if and only if L is onto.
Hints: THM 1: L is 1-1 iff ker(L)={0}.
THM 2: If dim V = n, then dim(ker(L)) + dim(range(L)) = n.

### Assignment #10 (due Wed. 5/30):

Section 4.4: 21, 22
Section 5.4: 1, 3, 5, 9
10A: Recall that, for a linear transformation L:V->W, we have that (range L) is a subspace of W.
Finish our proof from class by showing that (range L) is closed under scalar multiplication.

### Assignment #9 (due Tue. 5/29):

Section 4.2: 31, 33, 34, 36
Section 4.4: 23, 24, 25, 28
Section 4.5: 31
9A: Prove the following Corollary from class...
For a linear transformation L, if L(x) = v and L(y) = v, then x-y is in ker L.

### Assignment #8 (due Mon. 5/28):

Section 4.4: 13, 19, 25, 35
Section 4.7: 1, 3, 9, 13
8A: Let [u]_s denote the coordiante vector of u with respect to basis S.
Given vectors u,v and real scalar c, prove that
(i) [u+v]_s = [u]_s + [v]_s
(ii) [cu]_s = c[u]_s

### Midterm: Takehome, Due Friday, May 25

(Turn in HW#7 either Thursday or Friday)

### Assignment #7 (due Thu. 5/24 Fri. 5/25):

Section 1.7: 17, 21, 26, 31, 36[M]
Section 1.8: 3, 11, 16
Section 4.2: 30
Section 6.2: 29

### Assignment #6 (due Wed. 5/23):

Section 6.1: 19
Section 6.2: 12, 15, 33 (Consider A'A... what's going on here?)
Section 6.3: 1, 8, 19
Section 6.4: 10

5A: Write an m-file to define a new MATLAB function "gs(A)" that applies the Gram-Schmidt process to the columns of a matrix A. All the following work should be done in MATLAB.
Further instructions (necessary)
Excessive hints (optional)
gs.m (rough m-file template... recommneded)

```       |  1  2  3  0  |
A = |  3  0 -3  2  |
| -2  1  4 -1  |
```
(i) Apply your gs function to A. Let B=gs(A).
(ii) Verify that B's columns are orthonomal by computing B'*B. What should the result be, and why?
(iii) Verify that B's columns span the column space of A by solving Bx=v for each column v of A (use rref([B A(:,j)]) for each column j). Briefly explain the zero values in each such solution.
(iv) Use MATLAB, and specifically your gs function, to do Problem 6.4.22.

### Assignment #5 (due Tue. 5/22):

Section 6.1: 6, 12, 23, 28, 30
Section 6.2: 10, 26
Section 6.4: 2

### Assignment #4 (due Mon. 5/21):

In this assignment, you will learn to use MATLAB. All instructions for this assignment are contained in
hw4.gif.
Supplimentary material can be found in
hw4.m.
diary.txt.
Note that you may work together on all parts of this assignment. Please read the instructions in hw4.gif before beginning the assignment.

### Assignment #3 (due Fri. 5/18):

Section 5.1: 6, 15, 25, 29
Section 5.2: 6, 14
Section 5.3: 6, 10, 11
3A: Prove that the similarity relation is transitive, that is, if A ~ B and B ~ C then A ~ C.

### Assignment #2 (due Thu. 5/17):

Section 3.2: 11, 25
Section 4.1: 5, 7, 8, 32
Section 4.3: 11
2A: For the 4x5 matrix A from Problem 4.3.14 (pg 238), find bases for the null space, row space and column space of A. For each x in rs(A) and y in ns(A), verify that x and y are orthogonal (perpendicular).
2B: Find the eigenvalues of
```     | 1 -3 |
| 4  7 |```

### Assignment #1 (due Wed. 5/16):

Section 1.5: 14
Section 1.6: 28, 30
Section 2.1: 22 (Hint: Consider ABx)
Section 2.2: 4, 24
Section 3.1: 4, 9

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