Math 223, Section 101—Homework #4
due Friday, October 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. Friedberg, Insel, and Spence, Section 2.1, p. 74, #1, #2, #4, and #5
2. Friedberg, Insel, and Spence, Section 2.1, p. 75, #10 and #12
3. Friedberg, Insel, and Spence, Section 2.2, p. 84, #1, #2(a),(c),(d),(f),(g), and #3
4. Friedberg, Insel, and Spence, Section 2.2, pp. 84–85, #5(a)–(c),(e),(f) (replace F by R
throughout)
5. Friedberg, Insel, and Spence, Section 2.2, p. 85, #10
6. Friedberg, Insel, and Spence, Section 2.3, p. 96, #1 and #2
7. Friedberg, Insel, and Spence, Section 2.3, pp. 96–97, #3
8. Friedberg, Insel, and Spence, Section 2.3, p. 97, #4(a),(c), #6, and #7
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 2.2, pp. 84–85, #2(b),(e), #4, and #5(d)
2. (a) Using the matrices A, B, and C defined in Friedberg, Insel, and Spence, Section 2.3,
p. 96, #2(b), compute CA, BC t , and B t A. Which one of these is easy to check using
the answers given in the back of the book?
(b) Friedberg, Insel, and Spence, Section 2.3, pp. 96–97, #3(b). You can use the computation of [U ]γβ from the back of the book without having to justify it (although you
should understand why it’s correct!).
3. Consider the function T : P3 (R) → P (R) defined by T (p(x)) = xp(1 − x) − (1 − x)p(x).
T is a linear transformation (you don’t have to prove this).
(a) Find bases for N (T ) and R(T ).
(b) Compute the nullity and rank of T from your answers to part (a), and verify the Dimension Theorem for T .
(Hint: don’t be afraid to expand out a + b(1 − x) + c(1 − x)2 + d(1 − x)3 and so on.)
4. Let V and W be vector spaces, and let β = {β1 , . . . , βn ) be an ordered basis for V and
γ = {γ1 , . . . , γm } an ordered basis for W . Define a function G by G(T ) = [T ]γβ .
(a) What vector space is the domain of G? What vector space is the codomain of G?
(b) Prove that G is a linear transformation. (Hint: using one of the theorems from Section
2.2 will make this quite easy.)
(continued on next page)
5. The purpose of this problem is to make sure that we work through some potentially confusing notation. Reminders about the notation can be found in: Example 3 on page 43, the
Definitions on pages 80 and 82, the bottom of page 90 (after the Corollary), and Theorem
2.15 on page 93.
(a) Let {E 11 , E 12 , E 21 , E 22 } be the standard ordered basis for M2×2 (R), and define α =
{LE 11 , LE 12 , LE 21 , LE 22 }. Prove that α is a basis for L(R2 , R2 ).
(b) Let α be the ordered basis for L(R2 , R2 ) given in part (a), and let β be the standard
ordered bases for R2 . Let T : R2 → R2 be the linear transformation
T (a, b) = (a + b, 2b).
Write down [T ]α and [T ]β . (Hint: they won’t have the same shape.)
(c) Let T and β be as above. Calculate ([T ]β )4 . Use the answer to calculate T 4 (0, 1) .
(Recall that T 4 = T T T T denotes the fourfold composition of T with itself.)
6. Friedberg, Insel, and Spence, Section 2.3, p. 97, #11
7. Let V , W , and X be vector spaces, and let T : V → W and U : W → X be linear
transformations.
(a) Suppose that U T is onto. Prove that U is also onto.
(b) Suppose that U T is one-to-one. Prove that T is also one-to-one.
(c) Construct an example where U T is both one-to-one and onto, but U is not one-to-one
and T is not onto.
8. Let V be an n-dimensional vector space, and let β and γ be two ordered bases for V .
Suppose that T : V → V is a linear transformation such that [T ]γβ is the n × n identity
matrix In .
(a) Is T always the same as the identity linear transformation IV ? If so, why? If not, what
relationship between β and γ will cause T to equal IV ?
(b) Prove that no matter what β and γ are, T is always both one-to-one and onto. (Hint:
Theorem 2.5.)
Bonus question. Define two linear transformations L : R∞ → R∞ and R : R∞ → R∞ by
R (a1 , a2 , a3 , . . . ) = (0, a1 , a2 , . . . ) and L (a1 , a2 , a3 , . . . ) = (a2 , a3 , a4 , . . . ).
(You don’t have to verify that L and R are linear.) For any numbers r, ` ≥ 1, define Tr,` to be the
composition
· · L} .
Tr,` = Rr L` = R
· · R} L
| ·{z
| ·{z
r times
` times
(For example, T1,1 = RL and T35 = RRRLLLLL.) Prove that the infinite set
S = {Tr,` : r ≥ 1, ` ≥ 1}
is a linearly independent subset of L(R∞ ). (Hint: one stepping stone might be to prove that
Sr = {Tr,` : ` ≥ 1} is linearly independent for every fixed r ≥ 1.)