SPRING 2009
M341 Linear Algebra and Matrix Theory
Homework Assignments


Numbered problems are from the textbook. Other problems might be assigned occasionally. Grades are posted in UT Direct.

Homework #1 due 1/30

Read your class notes. Read the definition of parallel vectors p.7 and theorem 1.3 p.10.
1.1 p.13: 1(b), 2(b), 3(b), 4(b), 5(b), 6(b)(d), 7(b)(d)(f), 8(b), 11, 21, 23(a,d).

Homework #2 due 2/6

Read theorem 1.5 p.17, the definition of projection p.23, and theorem 1.10 p.24.
1.2 p.25: 1(b)(d), 3, 7, 8, 9, 10, 12, 14, 15(b), 16, 20.

Homework #3 due 2/13

Read:
  • Working "Backward" p.29
  • Result 4 p.31
  • Converse p.34
  • Proof by Contradiction p.35
  • Negating Statements p.38
    • 1.3 p.41: 1(a), 2, 4, 7, 8(b)(state contrapositive and converse only), 10(a,c), 19(b,d,f), 21.

    Homework #4 due Monday 2/23

    1.3 p.41: 13, 16.
    1.4 p.50: 1(d,h,j,l,n), 2(A,B,C,D,E,F,G), 3(b), 4, 5(b), 6, 8, 11(prove parts (2) and (3) of theorem 1.12), 12.
    1.5 p.60: 1(h,i,k,m,o), 2(b,e), 3(b), 9(b), 10(a)(i=[1,0,0], j=[0,1,0], k=[0,0,1]; use each vector as a column matrix), 14(b)(read (a) and (c) as well), 15, 18.

    Homework #5 due 3/6

    2.1 p.81: Solve the systems in: 1(b,f,h), 2(b,d); 5; 10.
    Read Theorem 2.1 p.80.
    2.2 p.90: 5(d), 12, 13.

    Homework #6 due 3/13

    Read "Inverses for 2x2 Matrices" p.110 and Theorem 2.15 and its proof p.114.
    2.3 p.103: 1(b), 4, 5(d), 9(b,d), 12, 18, 21.
    2.4 p.116: 1(a), 2(b), 3(b), 4(b,d), 5(b), 7(b)(look at thm 2.15), 9, 13, 16, 20.

    Homework #7 due 3/27

    Read: table 3.1 p.138, thm 3.11 p.145 and thm 3.13 p. 147.
    Show all your work.
    3.1 p.128: 1(b,d,f,h,j), 5(b), 9(b), 11(b), 15(b), 16.
    3.2 p.138: 2(b,d), 3(b), 4(b), 6, 7.
    3.3 p.148: 5, 6, 7, 8.

    Homework #8 due 4/10

    3.4 p.162: 1(b,d), 2(b), 3(d).
    4.1 p.178: 2, 6, 7, 12.

    Homework #9 due 4/17

    Read your class notes. You may use a calculator for the 4 basic operations.
    4.1 p.178: 14.
    4.2 p.187: 1(b,d,f,h,i,k), 2(b,d,f), 3(d), 4, 6, 16, 18 (ignore the "Do not forget ...").
    ** W is the subspace of R^4 consisting of all vectors of the form [a,b,a-b,a+c] where a, b, c are in R. Find a spanning set for W.
    ** Find a spanning set for the space of all 3x3 symmetric matrices.

    Homework #10 due 4/24

    4.3 p.198: 1(b), 5, 6, 7, 13, 14(a), 17, 18, 26. Hint for problems 6 and 7: in each case, describe span(S) as the solution set of a homogeneous system.

    Homework #11 due 4/29 This is due on Wednesday!

    4.4 p.208: 1(a,c,d), 2(c,d), 8, 10,

    Homework #12 due 5/8

    4.4 p.208: 5, 6, 11(b,d), 19(a), 27.
    4.5 p.221: 2, 4(a,b,d,e), 10(a), 14.
    4.6 p.231: 1(b), 4(b), 6(b), 7(b), 9(b).

    Recommended additional problems

    1.1 p.13: 9, 14(a), 15 (look at example 3 p.12), 23(b), 25, 26.
    1.2 p.25: 6, 11, 23.
    1.3 p.41: 9, 12, 17(I gave a different hint in class), 23, 25.

    Prove by induction: the sum of the first n numbers is n(n+1)/2 (i.e. 1+2+...+n=n(n+1)/2)

    Prove by induction the derivative formula: (xn)'= n xn-1 (you may use that the derivative of x is 1, and the product formula).

    Let A be the 2x2 matrix with entries a11=a12= a22=1, a21=0. Compute An for several values of n≥1.
    Conjecture a formula for An for any n≥1. Prove your result by induction.

    1.4 p.50: 14, 15.
    1.5 p.60: 7, 14, 17, 19, 24(a), 25, 27.
    2.1 p.81: 11. Also choose a couple of systems.
    2.2 p.90: 1, 14.
    2.3 p.103: 12, 14, 22.
    2.4 p.116: 10, 11, 21.
    3.1 p.128: 7, 8, 13, 14, 18.
    3.2 p.138: 12, 16.
    3.3 p.148: 13, 21, 22.
    4.1 p.178:
    Show that the solution set of a homogeneous system of linear equations in n variables is a vector space (use the usual addition and scalar multiplication in Rn).

    Show that the set of all 2x2 matrices that commute with the 2x2 matrix with entries a11=1, a12=0, a21=0, a22=2, with the usual addition and scalar multiplication of 2x2 matrices is a vector space.

    4.2 p.187: 11, 13, 17, 22.

    Show that the solution set of a homogeneous system of linear equations in n variables is a subspace of Rn.

    Show that the set of all 2x2 matrices that commute with the 2x2 matrix with entries a11=1, a12=0, a21=0, a22=2, is a subspace of Mn.

    4.3 p.198: 19, 22(c), 26, 28.
    4.4 p.208: 13, 15, 20, 28.
    4.5 p.221: 7, 8, 15(a,c), 23, 25.
    4.6 p.231: 11(a,b), 12, 17, 20.
    4.7 p.246: 1(a,e,f), 2(a,b,d), 3, 6, 7, 8, 9, 15(a-f).
    5.1 p.260: 1, 2, 4, 5, 8, 9, 12, 13, 21, 23, 30, 32.
    5.2 p.274: 2, 3(a,d).
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