Math 223, Section 101—Homework #3
due Wednesday, September 28, 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. Friedberg, Insel, and Spence, Section 1.6, pp. 53–54, #1(a)–(l)
2. Friedberg, Insel, and Spence, Section 1.6, p. 54, #2(a),(c),(e), #3(a),(c),(e), #4, #5, and #7
3. Friedberg, Insel, and Spence, Section 1.6, p. 55, #9, #10(a),(c), and #13. In #9, use R4
instead of the book’s F 4 .
4. Friedberg, Insel, and Spence, Section 1.6, p. 56, #19 and #24
5. Friedberg, Insel, and Spence, Section 1.6, p. 57, #26
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. Friedberg, Insel, and Spence, Section 1.6, p. 54, #2(b),(d) and #3(b),(d)
2. Friedberg, Insel, and Spence, Section 1.6, p. 55, #10(b),(d). Simplify your answers—that
is, write your answers in the form a0 + a1 x + a2 x2 + · · · .
3. Let n be a positive integer. Prove that the set of polynomials
{1, x + 2011, (x + 2011)2 , . . . , (x + 2011)n }
is a basis for Pn (R). (Is there a way to use the standard basis for Pn (R) to help?)
4. Friedberg, Insel, and Spence, Section 1.6, p. 55, #14, except use R5 rather than the book’s F 5 .
5. Let n be a positive integer. Define We to be the set of all even polynomials of degree
at most n (that is, the set of all polynomials of degree n that are even functions, such as
3x2 − 4), and define Wo to be the set of all odd polynomials of degree at most n (such
as −x3 + 12x). Both We and Wo are subspaces of Pn (R) (you don’t have to prove this).
Determine, with justification, the dimensions of We and Wo .
6. Friedberg, Insel, and Spence, Section 1.6, p. 56, #15 and #22. In #15, use Mn×n (R) rather
than the book’s Mn×n (F ).
7. Let D be a set with exactly n elements. Prove that the dimension of the vector space
F(D, R) equals n by writing down a basis for F(D, R) with exactly n elements.
(continued on next page)
8. Let V be a vector space, and suppose that {v1 , . . . , vn } is a finite subset of V that generates
V . Define
B = vj : vj ∈
/ span {v1 , . . . , vj−1 } ;
that is, B is the set of all those vj (where j ranges from 1 to n) that are not in the span of
the vi ’s that come before it. (To decide what to do with v1 : we interpret {v1 , . . . , vj−1 } as
the empty set when j = 1; recall that span(∅) = {0}.) For example, if n = 3 and v1 6= 0,
and v2 is a multiple of v1 but v3 isn’t, then B = {v1 , v3 }.
Prove that B is a basis for V . (Remark: this problem in particular gives a proof of
Theorem 1.9.)
Bonus question. Let V = F(R, R). If f : R → R is a function, we say that f is “dependable” if
f is constant outside of a finite set—that is, f is dependable if there exist a real number a and finite
list of real numbers x1 , . . . , xn such that f (x) = a whenever x does not equal any of x1 , . . . , xn .
Let W be the set of all dependable functions.
1. Show that W is a subspace of V .
2. Find, with proof, a basis for W .