Math 4326 LINEAR ALGEBRA Spring 2008
Dr. Duval Homework
Thursday, April 3–REVISED
Follow the separate general guidelines for Parts A,B,C. Be sure to include and label all
four standard parts (a), (b), (c), (d) of Part A in what you hand in.
Orthogonal Projections and Minimization Problems
pp. 111–116
A: Reading questions. Due by 2pm, Wed., 9 Apr.
1. Find U⊥for U= span((9,1,5)) in V=R3. Describe U⊥geometrically in this
case.
2. Verify Theorem 6.29 in the case of question 1 above.
3. Find PUvfor v= (1,2,3) and U= span((9,1,5)) in V=R3.
4. In the example starting on p. 114, approximating sin xby a 5th-degree polynomial,
explain how Rπ
−π|sin x−u(x)|2dx is minimized using the inner product 6.39 and
Proposition 6.36.
B: Warmup exercises. For you to present in class. Due by end of class Thu., 15 Apr.
Verify the five properties of PUlisted near the top of p. 113.
Ch. 6: Exercises 19, 21.
Generalized Eigenvectors (part I)
pp. 164–167
A: Reading questions. Due by 2pm, Wed., 16 Apr.
1. The text states, in the middle of p. 164, that the operator in 5.19 “does not have
enough eigenvectors for 8.2 to hold.” Explain carefully what that means in this
case.
2. Show how the example at the top of p. 165 matches the equation near the bottom
of p. 164, V= null(T−λ1I)dim V⊕ · · · ⊕ null(T−λmI)dim V.
3. The text claims, in the margin of p. 165, that “if (T−λI)jis not injective for
some positive integer j, then T−λI is not injective . . . ”. Verify this claim.
4. Verify Proposition 8.5 for the linear operator T∈ L(F4) given by T(z1, z2, z3, z4) =
(z1, z3, z4,0).
B: Warmup exercises. For you to present in class. Due by end of class Thu., 17 Apr.
Ch. 8: 1.