Math 24 Winter 2012
Linear Algebra - Homework

A handout on logical notation and formulas, sets, and proofs that don't require linear algebra material is available here.

Due Wednesday, March 7
Chapter 15 exercises #4, 7*, 12 (a only), 13 (for 12a only), 18 (p. 302-304).

* In #7, Bessel's inequality, the subscript i needs to be moved from B to A.
In Parseval's identity, you must make the additional assumption that the A_i form an orthonormal basis for V (this is not needed for Bessel's inequality).
Due Wednesday, February 29
Chapter 13 exercises #8, 9 (p. 225-226); Chapter 14 exercises #6 (a,b only), 13, 28 (m = 2 only), 32 (p 261-265).
Due Wednesday, February 22
Chapter 10 exercises #19, 33 (see 31 for terminology) (p. 154-155), Chapter 11 exercises #6*, 8, 16 (p. 179-181), and the problems below.
1. Give a condition on ker(T) and Im(T) that is equivalent to the matrix for T : R^n -> R^n being nilpotent of index 2 (i.e., its square is zero).
2. Let M be the 4x4 matrix with main diagonal 1, 1, 0, 0, and all zeros elsewhere. Let B be the ordered basis {(1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1)} for R^4. If T : R^4 -> R^4 is a linear transformation such that for any X in R^4, the product of M with the B-coordinates of X is the B-coordinates of T(X), find a matrix for T relative to the standard basis of R^4. What kind of transformation is T?

* For #6, some notes: the dimension of V and W need not be equal, and the general matrix form is intended to include the cases of no 1s and of 1s all the way to the bottom or right edge of the matrix.
Due Wednesday, February 8
Chapter 10 exercises #1 (parts 3 and 4 only), 2 (FG only), 3, 7, 15 (p. 151-153).
Due Wednesday, February 1
Chapter 8 exercises #2, 6, 11, 18, 24, 26* (p. 109-112).

* In #26, assume S is a linearly independent subset of V.
Due Wednesday, January 25
Chapter 6 exercises #4, 5 (for 4 only), 8, 10, 22 (p. 72-74).

Notes: in #10, the second sentence of the hint should begin "show that the result spans S+T." For #22, Fun(S, V) is defined in Chp 3 exercise #10 on p 31.

I suggest looking at Chapter 7 exercise #10 (p 83) but you do not need to write it up for submission.
Due Wednesday, January 18
Chapter 4 exercises #4, 11, 13, 16, 27 (p. 42-44).

Chapter 5 exercises #9, 17, 21* (p 54-55).

* For #21, show the result for 3 vectors in R2 instead of 4 vectors in R3.
Due Wednesday, January 11
Chapter 2 exercises #7, 9, 10, and 12 (p. 22-23), and the exercises below.
1. Let Axiom 4' read "For every vector A, there is a vector -A such that A + -A = 0." That is, it is Axiom 4 with the condition of uniqueness removed. Prove the following.
Claim: If a vector space is defined as in the textbook, section 2.1, but with Axiom 4' above in place of Axiom 4, each vector will still have a unique additive inverse.
2. Show that Axiom 2 holds for P2(R), the polynomials of degree at most 2 and real coefficients, with standard addition and scalar multiplication. See page 25 if you need a reminder of the definition.