Linear Algebra -- Hoffoss

Linear Algebra -- Fall 2005

Handouts (requires Acrobat Reader)

Homework Assignments

Some sketches to solutions to these problems are in the back of the book. But first, they are only sketches which need to be filled in, second, looking at them before you have completely exhausted your own ideas for a problem will rob you of the ability to learn from working on the problem, and third, copying these solutions constitues an honor code violation. Please be thoughtful about how you use the solutions in the back!

# Date Due Assignment

1 Wed 9/7

Read section 1.1.
Section 1.1 # 1, 5, 6, 16, 24, 25, 28
A. Prove that the vector / point b = (2, -5) can be written as a linear combination of the vectors v = (1, 4) and w = (-2, 2).
B. Prove that every vector / point in the plane can be written as a linear combination of the vectors v = (1, 4) and w = (-2, 2).
[Hint: to prove something is true for *every* vector in the plane, choose an arbitrary vector b = (b₁, b₂) in the plane, and show that this generic b can be written as a linear combination of v and w. Your scalars will have the values b₁ and b₂ in them.]
C. Prove that b = (1, 0, 1) cannot be written as a linear combination of v = (1, 2, 3) and w = (3, 4, -5)

2 Wed 9/14

Read Section 1.2 before Friday's class
Section 1.2 # 4, 5b, 7, 8, 14, 20, 21, 25, 29 (Answers to True/False questions always need accompanying explanation.)
Read Section 2.1 before Monday's class

3 Wed 9/21

Read Section 2.2 before Friday's class; Section 2.3 before Monday's class
Section 2.1 # 1, 2, 4, 6, 8, 13, 18, 20, 27, 28
(no proofs this section, but don't forget to write complete sentences .... for 28, choose from answers in parentheses)
Section 2.2 # 1, 2, 5, 7, 11, 14, 17, 19, 25
Section 2.3 # 1, 2, 6, 11

4 Wed 9/28

Read Section 2.4 and 2.5 before Friday's class; Section 3.1 before Monday's class
Think about formula 2B on page 47 -- we will use this later in the term
Section 2.3 # 13, 17, 19, 20, 27 (do not use values in back of book), 28
Section 2.4 # 1, 2, 6, 7, 8, 12, 14, 17, 18, 24 (for true/false, give explanations)
Section 2.5 # 5, 7, 8, 11, 15
A. Let A be a nonsingular matrix. Show that A^-1 is also nonsingular, and (A^-1)^-1 = A.
B. Prove Fact 5 from class. (Note 5 in the book)

5 Wed 10/05

Read Section 3.1 before Friday's class; Section 3.2 before Monday's class
Section 2.5 # 23, 26, 30
Section 2.7 # 3, 7
A. Prove that if A is nonsingular, then A^T is nonsingular, and (A^T)^-1 = (A^-1)^T
B. Extra Credit: Let A be a nonsingular nxn matrix. Use mathematical induction to prove that A^m is nonsingular and (A^m)^-1 = (A^-1)^m for all natural numbers m.
C. Prove that the inverse of a symmetric matrix is also symmetric. In other words, let A be a(n arbitrary) symmetric, invertible matrix. Prove that A^-1 is symmetric.
D. Prove: If R is any (arbitrary) m x n matrix, then R^TR is a square symmetric matrix.
Section 3.1 # 1, 2, 4, 7
Extra Practice: (not for portfolio)

Prove that if A and B are two matrices that are the right shape to be multiplied, then (AB)^T = B^TA^T.
There's now an entire sheet of proof practice problems on this website. Look at the top of the page for Proofs Practice 1. You should strive to do them all.

6 Wed 10/12

Read Section 3.3 before Friday's class; Section 3.4 before Monday's class
Section 3.1 # 10, 11, 12, 17, 18, 19, 22, 24, 26, 27, 29 (many are short answer -- don't panic!)
A. Let A be an mxn matrix. Prove that C(A) is a subspace of R^m. (Be sure you explain why it is a subset of R^m in your proof!)
B. Let A be an mxn matrix. Prove that N(A) is a subspace of Rⁿ. (Be sure you explain why it is a subset of Rⁿ in your proof!)
---------- Up to this line is all that will be due for this portfolio ---------
Extra Practice: (not for portfolio)

Keep working on Proofs Practice 1
Keep an eye out for Proofs Practice 2!

7 Wed 10/19

Read Section 3.3 before Friday's class; Section 3.4 before Monday's class (I guess we're a little behind.....)
Section 3.2 # 1, 2, 7, 9, 11a, 13, 15, 18, 21, 27 (important)
Section 3.3 # 2b, 6, 8B
Section 3.3 # 19, 22, 26
Section 3.4 1, 3, 7, 10, 11, 15, 17, 22, 24, 33
Extra Practice: (not for portfolio)

Keep working on Proofs Practice 1
Proofs Practice 2 won't be posted until after the exam -- sorry!

8 Fri 10/21

Exam I covers material through Section 3.4

9 Wed 10/26

Read Section 3.5 for Monday's class
Work Section 3.5 # 1, 5, 7, 9, 13, 17, 20, 22, 25 for problem session Wednesday
Section 3.5 is incredibly important. We will do MANY problems from this section. You should do even more problems from this section if you feel at all uncertain about the topics here.

10 Wed 11/2

Read Section 3.5 and 3.6
Work Section 3.5 # 24, 27, 31, 32, 35, 37, 41
Section 3.6 # 2, 5, 13, 15
Extra Practice, not to be turned in: Let A be an mxn matrix with rank r, and let R = reduced row echelon form of A. Consider all of the following spaces:

C(A)
C(R)
C(A^T)
C(R^T)
N(A)
N(R)
N(A^T)
N(R^T)
Which of the 8 spaces have the same dimension? Which of the 8 spaces are the same vector spaces (have the same vectors)? I'll compile a chart (download Subspace Chart from the top of the page): you guys come up with reasons why one space is the same as / is not the same as another, and I'll compile the chart for the class.

11 Wed 11/9

Section 4.1 # 3, 11, 13, 17, 22, 29
Extra Credit: Let V be a subspace of a vector space Z, and let W be the orthogonal complement of V. That is, let W = {w in Z : w is orthogonal to every vector in V}. Prove or disprove that W is a subspace of Z.

11 Fri 11/18

Section 5.1 # 2, 3, 6, 12, 13, 28
Section 6.1 # 2, 4, 9, 10, 16, 17, 19, 20

11 Fri 12/2

Read Section 7.1 for Monday 11/21, section 7.2 and 7.3 for rest of week.
Section 6.2 # 3, 8, 10 *, 20, 26, 35, 36
A. Suppose D₀=2, D₁=2, and D_n+1 = 2D_n + 3D_n-1. Find a solution to this difference equation; that is, find a formula for D_n which can be calculated directly from n rather than generated term by term.
Extra Credit: Suppose nxn matrices A and B can be diagonalized. Prove that they share the same eigenvector matrix S if and only if AB=BA.
Section 7.1 # 3, 6, 8, 18, 23
Section 7.2 # 1, 2, 5, 11, 12, 16, 20, 22, 30

11 Fri 12/9

Section 7.3 #

#	Date Due	Assignment
1	Wed 9/7	Read section 1.1. Section 1.1 # 1, 5, 6, 16, 24, 25, 28 A. Prove that the vector / point b = (2, -5) can be written as a linear combination of the vectors v = (1, 4) and w = (-2, 2). B. Prove that every vector / point in the plane can be written as a linear combination of the vectors v = (1, 4) and w = (-2, 2). [Hint: to prove something is true for every vector in the plane, choose an arbitrary vector b = (b₁, b₂) in the plane, and show that this generic b can be written as a linear combination of v and w. Your scalars will have the values b₁ and b₂ in them.] C. Prove that b = (1, 0, 1) cannot be written as a linear combination of v = (1, 2, 3) and w = (3, 4, -5)
2	Wed 9/14	Read Section 1.2 before Friday's class Section 1.2 # 4, 5b, 7, 8, 14, 20, 21, 25, 29 (Answers to True/False questions always need accompanying explanation.) Read Section 2.1 before Monday's class
3	Wed 9/21	Read Section 2.2 before Friday's class; Section 2.3 before Monday's class Section 2.1 # 1, 2, 4, 6, 8, 13, 18, 20, 27, 28 (no proofs this section, but don't forget to write complete sentences .... for 28, choose from answers in parentheses) Section 2.2 # 1, 2, 5, 7, 11, 14, 17, 19, 25 Section 2.3 # 1, 2, 6, 11
4	Wed 9/28	Read Section 2.4 and 2.5 before Friday's class; Section 3.1 before Monday's class Think about formula 2B on page 47 -- we will use this later in the term Section 2.3 # 13, 17, 19, 20, 27 (do not use values in back of book), 28 Section 2.4 # 1, 2, 6, 7, 8, 12, 14, 17, 18, 24 (for true/false, give explanations) Section 2.5 # 5, 7, 8, 11, 15 A. Let A be a nonsingular matrix. Show that A^-1 is also nonsingular, and (A^-1)^-1 = A. B. Prove Fact 5 from class. (Note 5 in the book)
5	Wed 10/05	Read Section 3.1 before Friday's class; Section 3.2 before Monday's class Section 2.5 # 23, 26, 30 Section 2.7 # 3, 7 A. Prove that if A is nonsingular, then A^T is nonsingular, and (A^T)^-1 = (A^-1)^T B. Extra Credit: Let A be a nonsingular nxn matrix. Use mathematical induction to prove that A^m is nonsingular and (A^m)^-1 = (A^-1)^m for all natural numbers m. C. Prove that the inverse of a symmetric matrix is also symmetric. In other words, let A be a(n arbitrary) symmetric, invertible matrix. Prove that A^-1 is symmetric. D. Prove: If R is any (arbitrary) m x n matrix, then R^TR is a square symmetric matrix. Section 3.1 # 1, 2, 4, 7 Extra Practice: (not for portfolio) Prove that if A and B are two matrices that are the right shape to be multiplied, then (AB)^T = B^TA^T. There's now an entire sheet of proof practice problems on this website. Look at the top of the page for Proofs Practice 1. You should strive to do them all.
6	Wed 10/12	Read Section 3.3 before Friday's class; Section 3.4 before Monday's class Section 3.1 # 10, 11, 12, 17, 18, 19, 22, 24, 26, 27, 29 (many are short answer -- don't panic!) A. Let A be an mxn matrix. Prove that C(A) is a subspace of R^m. (Be sure you explain why it is a subset of R^m in your proof!) B. Let A be an mxn matrix. Prove that N(A) is a subspace of Rⁿ. (Be sure you explain why it is a subset of Rⁿ in your proof!) ---------- Up to this line is all that will be due for this portfolio --------- Extra Practice: (not for portfolio) Keep working on Proofs Practice 1 Keep an eye out for Proofs Practice 2!
7	Wed 10/19	Read Section 3.3 before Friday's class; Section 3.4 before Monday's class (I guess we're a little behind.....) Section 3.2 # 1, 2, 7, 9, 11a, 13, 15, 18, 21, 27 (important) Section 3.3 # 2b, 6, 8B Section 3.3 # 19, 22, 26 Section 3.4 1, 3, 7, 10, 11, 15, 17, 22, 24, 33 Extra Practice: (not for portfolio) Keep working on Proofs Practice 1 Proofs Practice 2 won't be posted until after the exam -- sorry!
8	Fri 10/21	Exam I covers material through Section 3.4
9	Wed 10/26	Read Section 3.5 for Monday's class Work Section 3.5 # 1, 5, 7, 9, 13, 17, 20, 22, 25 for problem session Wednesday Section 3.5 is incredibly important. We will do MANY problems from this section. You should do even more problems from this section if you feel at all uncertain about the topics here.
10	Wed 11/2	Read Section 3.5 and 3.6 Work Section 3.5 # 24, 27, 31, 32, 35, 37, 41 Section 3.6 # 2, 5, 13, 15 Extra Practice, not to be turned in: Let A be an mxn matrix with rank r, and let R = reduced row echelon form of A. Consider all of the following spaces: C(A) C(R) C(A^T) C(R^T) N(A) N(R) N(A^T) N(R^T) Which of the 8 spaces have the same dimension? Which of the 8 spaces are the same vector spaces (have the same vectors)? I'll compile a chart (download Subspace Chart from the top of the page): you guys come up with reasons why one space is the same as / is not the same as another, and I'll compile the chart for the class.
11	Wed 11/9	Section 4.1 # 3, 11, 13, 17, 22, 29 Extra Credit: Let V be a subspace of a vector space Z, and let W be the orthogonal complement of V. That is, let W = {w in Z : w is orthogonal to every vector in V}. Prove or disprove that W is a subspace of Z.
11	Fri 11/18	Section 5.1 # 2, 3, 6, 12, 13, 28 Section 6.1 # 2, 4, 9, 10, 16, 17, 19, 20
11	Fri 12/2	Read Section 7.1 for Monday 11/21, section 7.2 and 7.3 for rest of week. Section 6.2 # 3, 8, 10 *, 20, 26, 35, 36 A. Suppose D₀=2, D₁=2, and D_n+1 = 2D_n + 3D_n-1. Find a solution to this difference equation; that is, find a formula for D_n which can be calculated directly from n rather than generated term by term. Extra Credit: Suppose nxn matrices A and B can be diagonalized. Prove that they share the same eigenvector matrix S if and only if AB=BA. Section 7.1 # 3, 6, 8, 18, 23 Section 7.2 # 1, 2, 5, 11, 12, 16, 20, 22, 30
11	Fri 12/9	Section 7.3 #

Linear Algebra -- Fall 2005

Handouts (requires Acrobat Reader)

Links

Homework Assignments