MAT 211 Homework Assignments

Fall 2005

IMPORTANT:
Answers to "True or False" and "Show that..." problems MUST include a full justification.

For example...
If you are given the statement "all odd integers are prime," first determine whether it is true or false. Then make the resulting claim, and give its justification. The following is acceptable:

CLAIM: Not all odd integers are prime. PROOF: Even though 9 is odd, it is not prime, since 9 = (3)(3).
(This is not the only possible answer -- you could use, say, 15 or 25 instead of 9. The point is that this is the level of precision expected in your answer.)

If you are given the statement "the product of two odd integers is odd," first determine whether it is true or false. Then make the resulting claim, and give its justification. The following is acceptable:

CLAIM: The product of any two odd integers is odd. PROOF: If m and n are odd integers, then m=2k+1 and n=2l+1 for some integers k and l, so mn = (2k+1)(2l+1) = 4kl+2k+2l+1 = 2(2kl+k+l)+1. Therefore mn is odd.

However, the following is NOT acceptable:

CLAIM: The product of any two odd integers is odd. PROOF: 5 and 7 are odd. (5)(7)=35, and 35 is odd.
(Since the claim concerns ANY pair of odd integers, not just 5 and 7, checking the statement for just one pair of odd integers does not constitute a valid answer.)

The only difference between a "True or False" problem and a "Show that..." problem is that with the latter, you are given a statement which is KNOWN to be true and then asked to justify it.

# Problems Due Date
1
 Section 1.1 -- 6,10,18,22,34, Section 1.2 -- 6,10,18,34,36, True Or False (pages 38-39) -- 2,6 9/7/05
2
 Section 1.3 -- 2,4,14,20,28,36,58, Section 2.1 -- 4,8,10,12,34,46, True Or False (page 39) -- 35, True Or False (page 97) -- 2 9/14/05
3
 Section 2.2 -- 6,12,14,20,32, Section 2.3 -- 10,18,30,34,40, True or False (pages 97-98) -- 8, 49 9/21/05
4
 Section 2.4 -- 6, 10, 16 (give a proof of the identity), 38, 44, 74, Section 3.1 -- 4,10,14,34,40 (give a proof of your claim), True or False (page 98) -- 46, True or False (page 150) -- 4 (you can use the the idea of dimension to formulate the right claim, but do not use it in the proof) 9/28/05
5
 Section 3.2 -- 2,6 (for both of these problems give a proof why the set in question is or is not a subspace),14,24,32,42,53, Section 3.3 -- 8,16,24,30,36,60,62, True or False (page 150) -- 4, 26 10/12/05
6
 Section 4.1 -- 2,4,6,10,14,20, 56 (For problems 2 through 20 prove that the given set is or is not a subspace. For problems 2 and 4 prove that the alleged basis is linearly independent and spans the given subspace. For problem 2, recall that for a polynomial P in one variable t and a real number a, P(a)=0 if and only if t-a is a factor of P.), Section 4.2 -- 6,10,18,20,34,52,66 (For problems 6 through 34 prove that the given function is or is not a linear transformation, and if it is linear prove that it is or is not an isomorphism.), True or False (pages 183-184) -- 6,16 (here T:V -> W is an isomorphism of vector spaces, where V and W are not necessarily the same, or even necessarily finite-dimensional) 10/19/05
7
 Section 4.3 -- 2,7,8,16,23,24,26,42,46,54,62, Section 5.1 -- 10,16,18,26,28,40,42 10/26/05
8
 Section 5.2 -- 4,8,10,14,16,20,32,34 Section 5.3 -- 2,8,10,16,20,34,40,52,56 (In section 5.3, for problems 2, 8, and 10 prove that the matrices in question are or are not necessarily orthogonal. Likewise, for problems 16 and 20, prove that the matrices in question are or are not necessarily symmetric. For problem 56, prove that L is or is not linear, and if it is linear prove that it is or is not an isomorphism.) True or False (pages 245-246) -- 27, 44 11/2/05
9
 Section 5.5 -- 4,6,10,12 (here the function is defined on the closed interval from -pi to pi),18,20,24 (for this problem, instead of letting the ambient space be polynomials with an integral inner product, let it be any inner product space whatsoever and work solely with the inner product axioms and the Gram-Schmidt process--this does not make the problem easier or harder, but it makes it more straightforward),26 (here the function is defined on the closed interval from -pi to pi), True or False (pages 245-246) -- 26, 28 11/9/05
10
 Section 6.1: 4,8,10,12,18,30, Section 6.2: 2,8,14,18,24,30, Section 6.3:14,16,24 11/23/05
11
 Section 7.2: 2,6,8,16,22,32,42,44, Section 7.3: 4,8,12,18,20,28,37, True or False (page 362): 6,26 11/30/05
12
 Section 7.4: 2,10,16,24,32 (note for this problem that you are asked for A to the t-th power, *not* the transpose of A), True or False (page 362): 34,36 12/7/05

"Mathematics is like your daughter Helena, who suspects every time a suitor appears that he is not really in love with her, but only interested in her because she is a princess. She wants a husband to love her for her own beauty, wit and charm, and not for the wealth and power which he can get by marrying her. Similarly, mathematics reveals its secrets only to those who approach it with pure love, for its own beauty. Those who do, are of course also rewarded with results of practical importance. But if someone asks at each step: 'what can I profit by this?' he will not get far. ...You remember I told you the Romans would never be really successful in applying mathematics. Well, now you see why: they are too practical minded." --attributed to Archimedes