Math 110 - Linear Algebra - Homework Assignments

Homework is due in section at the beginning of section the day it is due (Thursday). A few problems on each weekly homework assignment will be chosen at random to be graded. You are encouraged to work in groups in homework, but you must each turn in your own work. No late homework will be accepted, since answers will be available at Copy Central Northside and/or here the day after they are due. The lowest three homework grades will be dropped. The material in this class can only be learned by doing lots of problems, so the homework is very important.

According the textbook authors' errata list, unless the problem notes otherwise, assume that all fields used in examples and exercises have characteristic 0 (see Appendix C).

  • Assignment 1, due Sep 8 (last modified Aug 27, 9:40pm)
  • (1) Prove part (a) of the Cancellation Law: for all x, y, z in field F, x+z = y+z implies x = y
  • (2) Prove Z2 is a field. Hint: you can either make tables of all possible values needed to confirm the properties in the definition, or claim it as a special case of the next problem
  • (3) (Extra credit) Prove Zp is a field. Hint: To show there is a multiplicative inverse of any nonzero b, apply pigeon-hole principle to set of p-1 numbers 1*b mod p , 2*b mod p,..., (p-1)*b mod p
  • (4) Sec 1.2: 1 (justify your answers), 7, 9, 12 (but for odd functions, f(-t) = -f(t), not even), 15, 16, 19,
  • (5) Sec 1.3: 1 (justify your answers), 2d, 5, 8, 9, 11 (as stated, and also changed to read "f(x)=0 and f(x) has degree <= n"), 12, 15, 23, 28
  • (6) Sec 1.4: 1 (justify your answers), 13
  • (7) Sec 1.5: 1 (justify your answers), 12 (SEC 1.5 POSTPONED UNTIL HOMEWORK 2)
  • Answer to Assignment 1, in postscript or pdf. (last updated Sep 9, 10:00am)
  • Assignment 2, due Sep 15 (last modified Sep 9, 7:15am)
  • (1) Sec 1.5: 1 (justify) (postponed from last time), 2bd, 8, 9, 12 (postponed from last time), 13, 17
  • (2) Recall that that the set of symmetric nxn matrices form a subspace W of M_{n x n}(F). Find a basis of W. What is the dimension of W?
  • (3) Sec 1.6: 1 (justify), 5 (justify), 11, 12, 13, 29, 31
  • Answer to Assignment 2, in postscript or pdf. (last updated Sep 16, 2:50pm)
  • Assignment 3, due Sep 22 (last modified Sep 16, 5:20am)
  • Sec 2.1: 1 (justify your answers), 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 35, 38
  • Answer to Assignment 3, in postscript or pdf. (last updated Sep 23, 3:05pm)
  • Assignment 4, due Sep 29 (last modified Sep 23, 6:05am)
  • Sec 2.2: 1 (justify your answers), 3, 4, 6, 9, 10, 11, 14
  • Sec 2.2: 16 (also use this result to prove the Dimension Theorem, in the case where dim(V) = dim(W))
  • Sec 2.3: 1 (justify your answers), 3, 8, 12, 13
  • Answer to Assignment 4, in postscript or pdf. (last updated Sep 29, 9:25pm)
  • Assignment 5, do Practice Midterm 1
  • Answer to Assignment 5, in postscript or pdf. (last updated Oct 6, 3:20pm)
  • Assignment 6, due Oct 13 (last modified Oct 9, 8:25pm)
  • (1) Let G be a graph with n vertices, and A its n by n incidence matrix. Let B = sum_{k=1 to (n-1)} A^k. Let i be unequal to j, and suppose B_ij = 0. Prove that (A^m)_ij = 0 for all m > 1.
  • (2) Let G be a graph with n vertices and two kinds of edges: red edges and blue edges. Let R be the n by n incidence matrix of just of the red edges, and let B be the n by n incidence matrix of just the blue edges. Using R and B, express a matrix C with the property that C_ij = # paths from i to j consisting of one blue edge following by one red edge. Using R and B, express a matrix D with the property that D_ij = # paths from i to j consisting of one red edge following by one blue edge.
  • (3) Sec 2.3: 15
  • (4) Sec 2.4: 1 (justify your answers), 4, 5, 9, 10
  • (5) Sec 2.5: 1 (justify your answers), 6ab, 10
  • Answer to Assignment 6, in postscript or pdf. (last updated Oct 15, 8:45am)
  • Assignment 7, due Oct 20 (last modified Oct 15, 8:40am) click here
  • Questions 6, 7, 8 and 9 postposed until next week!
  • Answer to Assignment 7, in postscript or pdf. (last updated Oct 22, 4:25pm)
  • Assignment 8, due Oct 27 (last modified Oct 22, 7:25am). In addition to questions 6, 7, 8 and 9 from last time, click here.
  • Answer to Assignment 8, in postscript or pdf. (last updated Oct 27, 8:50pm)
  • Assignment 9, due Nov 3 (last modified Oct 28, 3:40pm): click here.
  • Answer to Assignment 9, in postscript or pdf. (last updated Nov 6, 5:45am, fixed bug in answer to last question)
  • Assignment 10, due Nov 10 (last modified Nov 3, 9:25pm): click here.
  • Answer to Assignment 10, in postscript or pdf. (last updated Nov 11, 4:00pm)
  • Assignment 11, due Dec 1 (posted Nov 21, 2:50pm, last modified Nov 21, 2:52pm): click here. Note that this assignment is longer than usual, since you have more time to work on it.
  • Answer to Assignment 11, in postscript or pdf. (last updated Dec 5, 4:50am)
  • Assignment 12, due Dec 8 (posted Dec 2, 6:55am) click here.
  • Answer to Assignment 12, in postscript or pdf. (last updated Dec 9, 11:30am)