## Homework assignments for Math 21a, Fall 2001

The red buttons will take you to selected solutions to the HW problems, posted after the homework has been graded. Generally speaking, these will be the ones that students had the most difficulty with. The solutions are in PDF format.

Assignment Due Date Reading Problems Solutions
1 Sept. 6 Section 1.1 6/7,8,14,15,18
2 Sept. 13 1.2
1.3
20/6,8,24,30,34
34/2,3,4,14,17,18

3 Sept. 20 1.3
2.1
34/22,28,32,47
46/2,3,6,40,41,43*
4 Wednesday
Sept. 26
2.1
2.2
46/8,9
61/1,4,10,13,19,20,22,42

5 Monday
Oct. 1
2.3 70/4,7,24,29,32,41,42
Write a well written proof
for problem A, given below.

6 Thursday
Oct. 11
2.4

3.1
85/5,6,15,16,17,42,43
Problem B, given below.
106/6,7,15,16

7 Oct. 18 3.1
3.2
106/23,27,32,34,35,38,49
118/4,6,12,14,22,23,35,40,45

8 Monday
Oct. 29
3.3
3.4
130/22,23,25,34,38,40,41,44
142/2,6,14,15,16,43,45,46

9 Nov. 1 5.1
5.2
190/4,10,12,15,16,22,23,26,30
199/4,6,30,32,35

10 Nov. 8 5.3
6.1
209/1,4,6,11,14,15,20, and problem C
251/3,8,18,29,34

11 Monday
Nov. 19
6.2 265/2,5,8,11,14,24,37
12 Nov. 29 6.3
7.1
283/2,9,10,11,19,20
300/1,2,3,4,6,8,9,15,16,20

13 Dec. 6 7.2
7.3
310/2,14,20,21,22
324/5,6,21,31,32,34,37

14 Monday
Dec. 10
7.4
8.1
337/4,5,14,33,34
372/3,4,8

*Note: for problems where you are asked to show that a function is linear, use the definition given in class (or in the book as "Fact" 2.2.1).

A. Finish the proof given in class that the inverse of a linear function is also a linear function, by proving:
Let T:RnRn be a linear function which is invertible, and let S be its inverse. Then for all x in Rn , and all real numbers r, we have S(rx) = rS(x).

B. Let T:RnRn be a linear function that is represented by a matrix A. Suppose that the matrix A is invertible, ie that there is a matrix B with AB = BA = In. Prove that T is one-to-one and onto. (This is the converse of what we proved in class.)

C. Prove that if a vector v in Rn satisfies v · w = 0 for all w in Rn, then v= 0. Combine this with the result of problem 1 (p. 209) to prove that for any matrix A, and any vectors v and w, we have (image(A))¬ = ker(AT). (The ¬ sign is supposed to be the upside down T denoting orthogonal complement, but I couldn't get it to some out on the web page.)

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