Math 223, Section 101—Homework #1
due Wednesday, September 14, 2011 at the beginning of class
Practice problems. Don’t turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to you to check your understanding of the material.
1. Make sure that you know the definitions of each of the following vector spaces, including
how addition and scalar multiplication work and which element is the zero element.
• Rn for any positive integer n
• R∞
• Mm×n (R) for any positive integers m and n
• Pn (R) for any positive integer n
• P (R)
• F(S, R) for any domain S
Also make sure you know the things that are true in every vector space, from Section 1.2 of
Friedberg, Insel, and Spence, and get used to reading “a field F ” in the book and changing
that in your mind to “the real numbers R”.
2. Friedberg, Insel, and Spence, Section 1.2, pp. 12–13, #1(a)–(d),(f)–(i),(k)
3. Friedberg, Insel, and Spence, Section 1.2, p. 13, #2, #3, and #4(a),(c),(e),(g)
4. Friedberg, Insel, and Spence, Section 1.2, p. 14, #8. Think explicitly about why every step
is true in any vector space, using the facts found in Section 1.2.
5. Friedberg, Insel, and Spence, Section 1.3, pp. 19–20, #1(b),(c),(e),(f)
6. Friedberg, Insel, and Spence, Section 1.3, p. 20, #2(a),(c),(e),(g), #3, #4, and #8(a),(c),(e)
7. Friedberg, Insel, and Spence, Section 1.3, p. 21, #15
8. Friedberg, Insel, and Spence, Section 1.4, p. 32, #1(a)–(c)
Homework problems. Write up and hand in solutions to these problems. Please have your solutions to these problems in the correct order on your pages. Write clearly and legibly, in complete
sentences; the solutions you hand in will be better if you think about how to best phrase your answer before you begin to write it. Please staple the pages together (unstapled solutions will not be
accepted).
1. A function f ∈ F(R, R) is called an even function if f (−t) = f (t) for every real number t.
A function g ∈ F(R, R) is called an odd function if g(−t) = −g(t) for every real number
t. Prove that the set of even functions and the set of odd functions are both subspaces of
F(R, R).
2. Friedberg, Insel, and Spence, Section 1.3, pp. 20–21, #5 and #20. You can use, without
having to prove them, the statements in Friedberg, Insel, and Spence, Section 1.3, p. 20, #3
and #4 if you wish.
3. Friedberg, Insel, and Spence, Section 1.3, p. 20, #8(b),(d),(f)
(continued on next page)
4. Prove that a subset W of a vector space V is a subspace of V if and only if the following
two conditions hold:
(d) W is nonempty;
(e) whenever a is a real number and x, y ∈ W , then ax + y ∈ W .
5. Let V be a vector space, and let x 6= 0 be a nonzero vector in V . Prove that span({x}) =
{ax : a ∈ R}.
6. Friedberg, Insel, and Spence, Section 1.4, p. 34, #12
7. Friedberg, Insel, and Spence, Section 1.4, p. 35, #15
8. (a) Let W be the set of all convergent sequences in R∞ , that is, the set of all sequences
{a1 , a2 , a3 , . . . } such that limn→∞ an exists. Prove that W is a subspace of R∞ .
(b) Let X be the set of all eventually zero√sequences in R∞ ; typical elements of X
are {3, 2, 1, 0, 0, 0, . . . } and {0, π, 0, e, − 2, 42, 0, 0, 0, . . . }, where all remaining elements equal 0. Prove that X is a subspace of the subspace W defined in part (a).
Bonus question. Of the following two definitions, one correctly defines a vector space and one
does not. For the correct definition, prove that all the axioms (VS 1)–(VS 8) are satisfied; for the
incorrect definition, show why at least one of the axioms is not satisfied.
(A) Let V1 = {(x, y) : x > 0, y > 0} be the set of all ordered pairs of positive real numbers.
Define vector addition on V1 by the formula
(x1 , y1 ) + (x2 , y2 ) = (x1 x2 , y1 y2 )
and define scalar multiplication on V1 by the formula
c(x, y) = (xc , y c ).
(B) Let V2 = [0, 360) = {x ∈ R : 0 ≤ x < 360}. Define vector addition on V2 by considering
the elements of V2 as angles measured in degrees; for example, 30 + 30 = 60 and 270 +
270 = 180. Define scalar multiplication in the same way; for example, 6 · 90 = 180 and
1
· 270 = 135. In other words, do addition and multiplication normally, but then reduce the
2
answer modulo 360.