Math 22: Homework Assignments

Homework will be assigned section-by-section but collected weekly, on Wednesdays. 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.

6.3 p 400 #3, 11, 15, 17, 19, 24

6.5 p 416 #5, 9, 13, 20, 23

6.6 p 425 #1, 3

**Assignment 8, Due Wednesday, August 13**

5.4 p 333 #3, 5, 13, 20, 21

6.1 p 382 #6, 12, 25, 28, 30

6.2 p 392 #7, 13, 17, 26, 30, 33

Proof (due Monday, August 11 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 W_1,W_2, ..., W_n be subspaces of R^n for some n. The *sum* of the W_{i} is

Σ W_i= {w_1+w_2+...+w_n such that w_i is in W_i for i between 1 and k}.

(ii) R^{n} is the *direct sum* of subspaces W_1, W_2, ..., W_n if their sum is R^{n} and each vector in R^n can be expressed uniquely as w_1+w_2+ ... + w_n, where w_i is an element of W_i.

**Theorem.** Let W_1, W_2, ..., W_n be subspaces of R^n that sum to R^n. Then R^n is the direct sum of W_1, W_2, ..., W_n if and only if the sum of the dimensions of the W_{i} is n.

**Assignment 7, Due Wednesday, August 6**

5.1 p 308 #2, 6, 12, 25, 31

5.2 p 317 # 2, 9, 15, 19

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.

No proof this week.

**Assignment 6, Due Wednesday, July 30**

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 Wednesday, July 23**

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 Wednesday, July 16**

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 c**u** 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 Wednesday, July 9**

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

Drill problems: several extra credit drill worksheets for sections 1.9, 2.1, and 2.2 are on Blackboard. Do as many or as few of these drills as you wish -- each one counts for extra credit.

No proof this week because of Exam I.

**Assignment 2, Due Wednesday, July 2**

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).

Proof assignment (reminder: keep this separate from the rest of your homework and make sure to write your name on it!)

Prove or disprove: A homogeneous system of two linear equations and three unknowns with two free variables in its coefficient matrix will have a solution space spanned by two linearly independent vectors.

BONUS CHALLENGE PROBLEM Prove or disprove: A homogeneous system of linear equations with k free variables in its coefficient matrix will have a solution space spanned by k linearly independent vectors.

**Assignment 1, Due Wednesday, June 25**

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 {**a**_{1}, **a**_{2}, **a**_{3}}.

Drill problems: the extra credit drill worksheets for sections 1.1 and 1.2 are on Blackboard (in the Documents section).

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.