## Linear AlgebraMath 4326

Spring 2016 Course Homepage
Prof. Bruce Peckham
Department of Mathematics and Statistics
University of Minnesota, Duluth

### Homework Assignments

Set 1, due FRI Jan. 15
• Lay, Lay and McDonald: Read Preface, Note to Students and Chapter 1, Sections 1.1-1.2. These sections should mostly be review of material from Math 3280.
• Sec. 1.1: Practice problems 1-4, Exercises 1,7,19,22,23,24,29. To turn in: 22, 24
• Sec. 1.2: Practice problems 1-3, Exercises 1,2,6,7,10,13,14,15,16,19-28. To turn in: 6,10,14,16b,18,20

Set 2, due FRI Jan. 22
• Read Chapter 1, Sections 1.3-1.5. These sections should mostly be review of material from Math 3280.
• Sec. 1.3: Practice problems 1-3, Exercises 8,12,14,18,20,22,24,25,26,27b. To turn in: All assigned "Exercises".
• Sec. 1.4: Practice problems 1-3, Exercises 1,3,5,7,9,14,15,17,19,23,26,27,30,31,32. To turn in: 14,15,17,19,26,27,30,31,32.
• Sec. 1.5: Practice problems 1-3, Exercises 1,5,7,10,12,13,29,30,31,32,33,35. To turn in: 10,30,31,32.

Set 3, due FRI Jan. 29
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• Read Chapter 1 and 2, Sections 1.7-1.9, and 2.1-2.2.
• Sec. 1.7: Practice problems 1-2, Exercises: 1,4,7,10,11,15,19,20,21,23,24,25,29,30, 32,33,36. To turn in: 4,10,20,24,30,32,36.
• Sec. 1.8: Practice problems 1-3, Exercises: 17,18,19,20,25,31. To turn in: 18,20,31.
• Sec. 1.9: Practice problems 1-2, Exercises: 2,4,15,18,34. To turn in: 4,18,34.
• Sec. 2.1: Practice problems 1-3, Exercises: 12,13,17,23,27. To turn in: 12,23.
• Sec. 2.2: Practice problems 1-3, Exercises: 1,5,11,13,15,17,31,35. To turn in: 11,15,31,35.

Set 4, due FRI Feb. 5
• Read Sections 2.3, 2.5 (LU Factorization only), 2.8,2.9.
• Sec. 2.3: Practice problems 1-3, Exercises: To turn in: 4,7,20,24,30,32,36, Extra Credit (turn in on a separate sheet to Prof. Peckham): 10.
• Sec. 2.5 Exercises: Turn in: 1, Extra Credit (turn in on a separate sheet to Prof. Peckham): 10.
• Sec. 2.8: Practice problems 1-3, Exercises: 2,7,10,11,13,15,18,20,22,23,26,28. To turn in: 2,10,18,20,26,28.
• Sec. 2.9: Practice problems 1-3, Exercises: 1,5,7,10,14,15,16,17. To turn in: 10,14,16. Extra Credit: 27.

Set 5, Due TUES. Feb 9.
• Sec. 3.1: Practice problem. Exercises: 1,10,19,20,21,26,27,30,33,36,37,42. To turn in: 10,20,30,36,42. (Do not quote resuts from Section 3.2.)
• Sec. 3.2: Practice problems 1-2. Exercises: 15,18,19,21,24,31,34,39. To turn in: 18,24,34.

Test 1 Thursday Feb 11 5-6:30pm. Room: MonH 70.
Covers all sections assigned in Chapters 1,2 and 3. Expect a mix between computational and conceptual problems. Many will be similar to HW problems, including those assigned but not turned in. A list of topics and possible proofs is here.

Set 6, Due Fri Feb 19.
• Read Sections 4.1, 4.2, 4.3, 4.4
• Sec. 4.1: Practice problems 1-3. Exercises: 1,5-9,16,18,23a-d. To turn in: 6,7,8,18.
• Sec. 4.2: Practice problems 1-3. Exercises: 1,4,7,16,17,20,23. To turn in: 4,16,20,23.
• Sec. 4.3: Practice problems 1-4. Exercises: 1,2,8,11,12,14,16,20,21,22,23,24,25,31,34. To turn in: 8,12,14,20,24,34.

Quiz 1, Fri. Feb. 19, last 10 min.: Prove that a given subset of R^n (for a specific n) is a subspace of R^n. Quiz1Solutions.
Set 7, Due Fri Feb 26.
• Read Sections 4.4, 4.5, 4.6.
• Sec. 4.4: Practice problems 1-2. Exercises: 2,5,8,9,12,13,15,17,19,28,32. To turn in: 2,8,12,19,32.
• Sec. 4.5: Practice problems 1-2. Exercises: 1,6,9,10,15,16,18,19,25,26. To turn in: 6,10,16,18,26.
• Sec. 4.6: Practice problems 1-4. Exercises: 1,4,5,6,7,8,11,14,16,18,20,21,25. To turn in: 4,6,8,14,16,18,25.

Quiz 2, Fri. Feb. 26, last 10 min.: One question on coordinates (like Sec 4.4 5,9 or 13), and one question to provide an example of a matrix with some given information like its rank, nullity, or number of columns or rows.
Set 8, Due Fri March 4.
• Read Sections 4.7, 5.1, 5.2.
• Sec. 4.7: Practice problems 1-2. Exercises: 2,6,7,10,13. Extra Credit: 20. To turn in: 2,6,10.
• Sec. 5.1: Practice problems 1-4. Exercises: 2,5,9,16,17,20,21,24,25,26. To turn in: 2,16,20,24,25,26.
• Sec. 5.2: Practice problem. Exercises: 4,7,9,14,15,18,21,24. To turn in: 4,14,18,24.

Quiz 3, Fri. March 4, last 10 min.: Two questions: 1. Prove that a span of any set of vectors in V is a vector subspace of V. (See examples 1 and 3 in Sec. 2.8.) 2. Find a change of basis matrix between two bases for either R^n or some other abstract vector space. (See Exercises 10-14 in Section 4.7.) Answer Key is here.

Spring Break

Set 9, Due Fri March 18.
• Read Sections 5.3, 5.4, 5.5.
• Sec. 5.3: Practice problems 1-3. Exercises: 2,3,6,7,14,20. To turn in: 2,6,14,20.
• Sec. 5.4: Practice problems 1-2. Exercises: 1,3,4,6,7,10,11,14,20,23,25. To turn in: 4,6,10,14,20,23,25.
• Sec. 5.5: Practice problem. Exercises: 1,6,7,12,13,18,23. To turn in: 6,12,18,23. Extra Credit: 24.

Quiz 4, Fri. March 18 (moved to Mon. March 21), last 10 min.: Repeat of Quiz 3. Some numbers may change.

Test 2 Tuesday March 22 (moved to Thursday March 24) 5-6:30pm. Room: LSci 185.
Covers sections 4.1-4.7 and 5.1-5.5. From sec. 5.5, you only need to know how to compute eigenvalues and eigenvectors when they are complex. Expect a mix between computational and conceptual problems. Many will be similar to HW problems, including those assigned but not turned in. A list of topics and possible proofs will be provided. is here.

Set 10, Due Fri April 1.
• Read Sections 6.1, 6.2, 6.3
• Sec. 6.1: Practice problems 1-3. Exercises: 3,8,11,15,18,19,30. To turn in: 8,18,30.
• Sec. 6.2: Practice problems 1-4. Exercises: 1,6,7,10,11,14,15,17,22,23,26,30. To turn in: 6,10,14,22,26.
• Sec. 6.3: Practice problems 1-2. Exercises: 1,3,6,7,10,12,13,18,23. To turn in: 6,10,12,18,23.

Quiz 5, Fri. April 1, last 10 min.: Possible topics: 1. Proof that the nullspace of an m by n matrix is a vector subspace of R^n, 2. Proof of the law of cosines (assuming the Pythagorean theorem), 3. Proof that vec(a) ``dot'' vec(b) = a b cos (theta), assuming the law of cosines, 4. compute by hand the projection of a vector onto a subspace, or onto its orthogonal complement.
Set 11, Due Fri April 8.
• Read Sections 6.4 (QR factorization optional), 6.5 (through Example 4 only), 6.7 (Two inequalities section optional)
• Sec. 6.4: Practice problems 1-2. Exercises: 4,8,12,17,22. To turn in: 4,8,12,22. EC: 16.
• Sec. 6.5: Practice problems 1-2. Exercises: 1,4,6,8,9,12,17. To turn in: 4,6,8,12.
• Sec. 6.7: Practice problems 1-2. Exercises: 1,4,6,8,25. To turn in: 1,4,6,8,25.

Quiz 6, Fri. April 8, last 10 min.: Possible topics: 1. For a subspace W of a vector space R^n: prove that the set of vectors perpendicular to every vector in W (that is, "W perp") is a subspace of R^n; 2. Compute the projection of a vector in R^3 to a 2-dimensional subspace W given a basis, but not necessarily an orthhogonal basis, for W; 3. Compute the inner of two vectors given a specific space and inner product (other than R^n with the dot product) or use the inner product compute a projection onto a subspace. Answer Key is here.
Set 12, Due Fri April 15.
• Read Sections 7.1 through Theorem 3, 7.2, 7.3 Theorem 6 only, 7.4 pp 416-419, and the conclusion of the Invertible Matrix Theorem on p. 423. The singular value decomposition from p. 419 on is NOT required.
• Sec. 7.1: Practice problems 1-2. Exercises: 5,6,7,10,12,13,19,24,25. To turn in: 6,10,12,24.
• Sec. 7.2: Practice problem. Exercises: 1,6,7,11,14,21,23,25. To turn in: 6,7,14,23,25.
• Sec. 7.4: Exercises: 1,4,14,17,20. To turn in: 1,4,14,17,20.

Quiz 7, Fri. April 15, last 10 min. Possible topics: 1. Orthogonally diagonalize a given symmetric matrix. 2.Given a quadratic form in two variables, write it as "x transpose A x"; make a change of variables x=Py so that the quadratic form in y has no y_1 y_2 terms. 3. Possible proofs: A. Show A^T A is symmetric. B. Show that if A is orthogonally diagonalizable (ie, if A=PDP^T where P^T = P inverse), then A is symmetric. (bottom of p. 398) C.Show that if A is symmetric, and v and w are eigenvectors of A for different eigenvalues, then v and w are orthogonal. (Thm 1, p. 397) D. Show the eigenvalues of A^T A are real and nonnegative. (top of p. 418, eqn (2)) Answer Key is here. Note that there are several correct ways to do problem 3b.

Test 3 Thursday April 21. Room: MWAH 195. 5:15 - 6:45pm
Covers sections 5.5, 6.1 - 6.5, 6.7, 7.1, 7.2, 7.4. List of topics is here.

Set 13 Application sections. Due Friday April 29.