Math 22: Homework Assignments

Homework will be assigned section-by-section but collected weekly, on Fridays. I advise you to do it as we cover the sections as it might be a little overwhelming piled up at the end of the week. Note also that showing work is very important because I will be giving a good percentage of problems whose numerical answer is in the back of the book.

A common complaint of students when assigned problems whose answers are not in the back of the book is that they have to do the problems without knowing how they are supposed to turn out. To solve this difficulty (beyond the role of the examples in the book and lecture), your text includes practice problems at the beginning of each section's exercises which are fully solved just before the next section. Take advantage of those as intermediates between examples (where you see the solution right with the problem) and odd-numbered exercises (where you have the answer but not the process).

Suggested Problems Post-Assignment 8

Not for handing in.

6.2 p 392 #7, 13, 17, 26, 30, 33
6.3 p 400 #3, 11, 15, 17, 19, 24
6.5 p 416 #5, 9, 13, 20, 23
6.6 p 425 #1, 3

Answers to the four even-numbered problems.

Assignment 8, Due Friday, May 25

5.3 p 325 #6, 16, 23, 24, 25, 26
#6 does not require computation. For #16 the eigenvalues are given in the note preceding exercise 7; you still have to determine the multiplicities but this gives you a leg up on factoring the characteristic function.
5.4 p 333 #3, 5, 13, 20, 21
6.1 p 382 #6, 12, 25, 28, 30

Proof (due Wednesday, May 23 to allow for rewriting time):

Recall that the sum of two vector spaces is the collection of all sums of vectors in those spaces. We make a more general definition:

Definition. (i) Let W1, W2, ..., Wk be subspaces of Rn for some n. The sum of the Wi is

Σ Wi = {w1+w2+...+wk : wi in Wi for i between 1 and k}.
(ii) Rn is the direct sum of subspaces W1, W2, ..., Wk if their sum is Rn and their pairwise intersection is the zero vector (i.e., the intersection of Wi and Wj is {0} for all i not equal to j).

Theorem. Let W1, W2, ..., Wk be subspaces of Rn that sum to Rn. Then Rn is the direct sum of W1, W2, ..., Wk if and only if the sum of the dimensions of the Wi is n.

Assignment 7, Due Friday, May 18

5.1 p 308 #2, 6, 12, 25, 31
5.2 p 317 # 2, 9, 15, 19

No proof this week.

Assignment 6, Due Friday, May 11

4.5 p 260 #9, 13, 24, 26
4.6 p 269 #2, 7, 10, 19, 30
4.7 p 276 #2, 4, 7, 13
4.9 p 296 #3, 6, 19

No proof this week because of Exam II.

Assignment 5, Due Friday, May 4

4.2 p 234 #6, 7, 10, 27, 28, 31
4.3 p 243 #2, 6, 14, 19, 33
4.4 p 253 #5, 13, 18, 20, 21, 32

Proof assignment:

Theorem. Let V and W be vector spaces, and T: V->W a linear transformation. Then
(i) If U is a subspace of V, and we set T(U) = {y in W: for some x in U, T(x) = y}, then T(U) is a subspace of W.
(ii) If Z is a subspace of W, and we set X = {x in V : T(x) in Z}, then X is a subspace of V.

Assignment 4, Due Friday, April 27

Chapter 3 assignment in pdf.
4.1 p 223 #8, 12, 21, 31, 32, 33, 34

Proof assignment: prove the following theorem. Note that as it is an "if and only if" statement, you will have to show each condition implies the other (this will be easier separate than trying to do both directions at once).

Definition. A subspace of a vector space V is a subset W such that 0 is in W and W is closed under vector addition and scalar multiplication (i.e., if u and v are in W and c is a scalar, u+v and cu are also in W).

Theorem. Let V be a vector space and W a subset of V. W is a subspace of V if and only if W is nonempty and closed under vector addition and scalar multiplication.

Assignment 3, Due Friday, April 20

1.9 p 90 #3, 5, 11, 18, 31, 32
2.1 p 116 #2, 10, 12, 18, 19, 20
2.2 p 126 #7, 11, 12, 14, 30, 31
2.3 p 132 #17, 18, 27, 33, 39

No proof this week because of Exam I.

Assignment 2, Due Friday, April 13

1.4 p 47 #6, 8, 13, 18, 20, 26
Note that for #6, 8 you aren't solving the system, just rewriting it.
1.5 p 55 #8, 9, 17, 21, 25, 26
For #25 and 26 (not 21, there's nothing in 21 you could really be formal about) you don't need to be formal.
1.7 p 71 #2, 15-20, 36, 37, 38
For 15-20 you do not need any computation. If you've really internalized the ideas of linear independence, only two of the six should take you more than a glance, and those will take a very small amount of mental arithmetic.
1.8 p 79 #9, 16, 20, 24, 31, 32, 34
Originally I also had #7, 8, 11, 29, and 30 in the assignment, so it would be good to look at those in particular (though you are responsible for all the material, not just the exact sorts of questions I give you for homework).

No proof this week, but rewrites of previous proof due.

Assignment 1, Due Friday, April 6

1.1 p 11 #7, 15, 16, 17, 25
For #17 remember to explain.
1.2 p 25 #2, 8, 10, 16, 19, 23-26
In 23-26 be brief, these aren't formal proofs; you may find 25 useful for answering 26.
1.3 p 37 #10, 13, 16, 20, 21, 25
In 20 be specific ("a plane" is not enough); in 25(a) note they are not asking about the span of {a1, a2, a3}.

Proof Assignment (keep separate from homework)
Essentially, 1.2 #29, but a little more formal (see below). Hint: what gives a cap to the number of pivots, rows or columns?

Definition: A system of linear equations is underdetermined if it has fewer equations than unknowns.

Theorem: Any consistent underdetermined system of linear equations has an infinite solution set.

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