Linear Algebra (Math 232 A & B, Fall 2004)

Textbook

Textbook Distribution and Update Site

Course Syllabi

Section A (10:00): Syllabus [PDF format]
Section B (12:00): Syllabus [PDF format]

Books on Proofs

The books below are in the spirit of the recommended text "Nuts and Bolts of Proof." If you find that book helpful, you might consider ordering one or more of these from someplace like Amazon.com or Barnes & Noble (these are listed in order of my familiarity with them, which may not be the same order I would recommend them in).

Links

Old Exams (Fall 2003)

[Exam 1]  [Exam 2]  [Exam 3]  [Exam 4]  [Exam 5]  [Exam 6]

Homework exercises (updated Nov 16, 2004)
These are suggested exercises from Johnson/Riess/Arnold, 4th Edition.  Generally, the same problems have the same numbers in the 5th Edition.  Where this is not the case, a slash precedes the number for the 5th Edition.  Also, for the 5th Edition increase the chapter number by 1 for each problem from all but Chapter 1 of the 4th Edition.  So, for example, 2.4.34/36 would be 2.4.34 in the 4th Edition and 3.4.36 in the 5th Edition.  Is that clear?

Section
 Computational
 Theoretical
WILA
 1.1.1, 1.1.2, 1.1.11, 1.1.14, 1.1.42
SSSLE
 1.1.8, 1.2.49, 1.2.50, 1.2.53
 1.1.38/39
RREF
 1.1.27, 1.1.31, 1.1.34, 1.2.3, 1.2, 5, 1.2.8, 1.2.13, 1.2.15, 1.2.17, 1.2.21, 1.2.23, 1.2.27, 1.2.29, 1.2.31, 1.2.47
TSS
 1.2.38, 1.3.1, 1.3.3, 1.3.5, 1.3.6
HSE
 1.3.7-19 odd, 1.3.21, 2.3.25
NSM
 1.7.16, 1.7.17, 1.7.18, 1.7.21, 1.7.22, 1.7.24
 2.3.51
VO
 1.5.7, 1.5.15
LC

SS
 1.5.45, 2.3.15, 2.3.17, 2.3.19, 2.3.21
LI
 1.7.1-13 odd, 1.7.30, 1.7.41, 1.7.43, 2.4.1, 2.4.3, 2.4.7, 2.4.9, 2.4.27, 2.4.33
 1.7.49/51
MO
 1.5.1, 1.5.3
 1.6.44, 1.6.47
RM
 2.3.27-35 odd, 2.3.39, 2.3.41, 2.3.47, 2.4.19
 2.3.50, 2.3.52
RSM
 2.4.11, 2.4.13, 2.4.23
MM
 1.5.11, 1.5.23, 1.5.31, 1.5.33-35, 1.5.40, 1.5.55, 1.5.63, 1.6.1, 1.6.3, 1.6.5, 1.6.17, 1.6.26, 1.6.27, 1.6.30-32, 4.3.28, 4.3.29
 1.5.59, 1.5.60, 1.5.67, 1.7.47/49, 1.7.50/52, 1.7.51/53, 2.3.51
MISLE
 1.9.3, 1.9.7, 1.9.19, 1.9.23, 1.9.29/31, 1.9.37/39, 1.9.39/41
 1.9.52/54, 19.53/55, 1.9.54/56
MINSM
 3.7.15, 3.7.17, 3.7.30
 1.9.66/68
O
 1.6.21, 2.6.3, 2.6.5, 2.6.9, 2.6.12, 2.6.13
 1.6.46, 1.9.56/58, 2.6.22, 2.6.25, 2.6.28
VS
 2.2.27, 4.2.1, 4.2.2, 4.2.3, 4.2.5, 4.2.9, 4.2.11, 4.2.13, 4.2.15, 4.2.18, 4.2.19
S
 2.2.3, 2.2.5, 2.2.7, 2.2.15, 2.2.17, 4.3.1, 4.3.3, 4.3.5, 4.3.7, 4.3.9, 4.3.13, 4.3.17, 4.3.19, 4.3.23
 2.2.18, 2.2.21, 2.2.30, 2.2.31, 2.2.32, 4.3.30, 4.5.2
B
 4.3.27, 4.3.32, 4.4.1, 4.4.3, 4.4.5, 4.4.7, 4.4.14, 4.4.15, 4.4.17, 4.4.19, 4.4.21, 4.4.24
 4.4.32, 4.4.36, 4.4.37, 4.4.38, 2.5.31
D
 2.5.17, 2.5.23, 2.5.25, 2.5.27, 2.5.29, 4.5.1, 4.5.4, 4.5.5, 4.5.7
 2.5.38, 2.5.40, 4.5.17
PD
 2.5.7, 2.5.8, 2.5.9, 4.5.9, 4.5.11, 4.5.13
 2.5.30, 2.5.32, 2.5.36
DM
 3.2.1-3.2.4, 3.2.7, 3.2.9, 3.2.11, 3.2.17, 3.2.18, 3.2.19
 3.2.23, 3.2.24, 3.2.33, 3.2.34
EE
 3.1.3, 3.1.5, 3.1.7, 3.1.9, 3.1.15, 3.4.3, 3.4.5, 3.4.7, 3.4.9, 3.4.13, 3.4.21, 3.5.3, 3.5.5, 3.5.7, 3.5.9, 3.5.13, 3.5.17, 3.6.21, 3.6.23, 3.6.33 3.1.17, 3.1.19
PEE
 3.5.19
 3.4.25, 3.4.30, 3.5.21, 3.5.22, 3.5.23, 3.5.24, 3.5.25, 3.5.28, 3.5.29, 3.6.36, 3.6.37, 3.6.41
SD
 3.5.3, 3.5.5, 3.5.7
 3.5.25, 3.5.26, 3.5.27, 3.5.29, 3.5.43
LT
 2.7.1, 2.7.2, 2.7.5, 2.7.7, 2.7.11, 2.7.13, 2.7.15, 2.7.17, 2.7.19, 4.7.5, 4.7.7, 4.7.9, 4.8.1-4.8.4
 2.7.33, 2.7.38
ILT
 2.7.3
 4.7.20, 4.7.26
SLT
4.7.22
ILT/SLT
 2.7.29, 4.7.13, 4.7.16, 4.7.17, 4.8.5, 4.8.6
 4.7.18, 4.7.19, 4.7.21
IVLT
 4.8.7, 4.8.9, 4.8.11
 4.8.18-28
VR
4.5.18
MR
 4.9.1-10, 4.9-13, 4.9.14-16, 4.9.19
 4.9.28, 4.9.30
CB
 4.10.1, 4.10.3, 4.10.6, 4.10.9, 4.10.10, 4.10.11, 4.10.15, 4.10.16 4.10.17, , 4.10.18, 4.10.19, 4.10.20

Homework exercises (updated Sept. 28, 2004)

These are suggested exercises from Jim Hefferon's free linear algebra text, Linear Algebra. 

Section
 Section
 Computational
 Theoretical
WILA


SSSLE


RREF


TSS


HSE


NSM


VO


LC


SS
 2.I.2
 2.22a
 2.39, 2.41
LI
 2.II.1
 1.18
 1.24, 1.25, 1.26, 1.33, 1.38
MO
 3.IV.1
 1.7
 1.9, 1.13, 1.15a
RM


RSM


MM


MISLE


MINSM


VS


S
 2.I.2
 2.20, 2.21, 2.25, 2.26, 2.35
B
 2.III.1
 1.16-1.24, 1.28, 1.32, 1.34
 1.27, 1.30, 1.31, 1.33
D


PD
 2.III.2
 Any (2.14-2.34)

Reading Questions

After reading each section, send me an email to the address announced in class (not my UPS(dot)EDU account) with your answers to each of the three questions. Each answer will be graded as one point, there will be no partial credit. I will reply with a list of the questions you got credit for. Observe the following to ensure your answers are received and graded properly.

  1. Make your subject line exactly, exactly as follows: Math 232 X, where X is the upper-case acronym for the relevant section. So, for example, the first reading assignment answers would have the subject line (exactly): 
    Math 232 WILA
  2. Put your full name as the first line of the body of your message.
  3. Answer the questions in order.
  4. Answers are due at 10:00 in the evening prior to the day we begin discussing each section. They will not be accepted late.

Quick Links

Chapter SLE: [WILA]  [SSSLE]  [RREF]  [TSS]  [HSE]  [NSM]
Chapter V:  [VO]  [LC]  [SS]  [LI]
Chapter M: [MO]  [ROM]  [RSOM]  [MOM]  [MISLE]  [MINSM]
Chapter VS:  [VS]  [S]
Reading questions are continued in a PDF file [Reading Questions]

Section WILA

  1. Is the equation x2 + xy +tan(y3)=0 linear or not?  Why or why not?
  2. Find all solutions to the system of two linear equations 2x+3y=-8, x-y=6.
  3. Explain the importance of the procedures described in the  trail mix application (Subsection WILA.A) from the point-of-view of the production manager.

Section SSSLE

  1. How many solutions does the system of equations  3x + 2y = 4, 6x + 4y = 8  have? Explain your answer.
  2. How many solutions does the system of equations  3x + 2y = 4, 6x + 4y = -2  have? Explain your answer.
  3. What do we mean when we say mathematics is a language?

Section RREF

  1. Is the matrix below in reduced row-echelon form?  Why or why not? 
    1 5 0 6 8
    0 0 1 2 0
    0 0 0 0 1
  2. Use row operations to convert the matrix below to reduced row-echelon form. 
     2 1  8
    -1 1 -1
    -2 5  4
  3. Find all the solutions to the system below by using an augmented matrix and row operations.  Report your final matrix and the set of solutions.
    2x1 + 3x2 - x3 = 0
    x1 + 2x2 + x3 = 3
    x1 + 3x2 + 3x3 = 7

Section TSS

  1. How do we recognize when a system of linear equations is inconsistent?
  2. Suppose we have converted the augmented matrix of a system of equations into reduced row-echelon form.  How do we then identify the dependent and independent (free) variables?
  3. What are the possible solution sets for a system of linear equations?

Section HSE

  1. What is always true of the solution set for a homogenous system of equations?
  2. Suppose a homogenous sytem of equations has 13 variables and 8 equations.  How many solutions will it have?  Why?
  3. Describe in words (not symbols) the null space of a matrix.

Section NSM

  1. What is the definition of a nonsingular matrix?
  2. What is the easiest way to recognize a nonsingular matrix?
  3. Suppose we have a system of equations and its coefficient matrix is nonsingular.  What can you say about the solution set for this system?

Section VO

  1. Where have you seen vectors used before in other courses?  How were they different?
  2. In words, when are two vectors equal?
  3. Perform the following computation with vector operations 
      |1|        |7|
    2 |5| + (-3) |6|
      |0|        |5|

Section LC

  1. Earlier, a reading question asked you to solve the system of equations
    2x1 + 3x2 - x3 = 0
    x1 + 2x2 + x3 = 3
    x1 + 3x2 + 3x3 = 7
    Use a linear combination to rewrite this system of equations as a vector equality.
  2. Find a linear combination of the vectors 
    | 1| |2| |-1|
    | 3|, |0|, | 3|
    |-1| |4|, |-5|
    which equals the vector 
    | 1|
    |-9|
    |11|
  3. Use the same three vectors as in the previous question and build a linear combination that equals 
    | 2|
    | 5|
    |-2|

Section SS

  1. The matrix below is the augmented matrix of a system of equations, row-reduced to reduced row-echelon form.  Write the vector form of the solutions to the system. 
     1  3  0  6  0  9
     0  0  1 -2  0 -8
     0  0  0  0  1  3
  2. Let  S  be the set of  three vectors below.  
    | 1|   | 3|   | 4|
    | 2|   |-4|   |-2|
    |-1|   | 2|   | 1|
    Let   W  be the span of  S. Is the vector 
    |-1|
    | 8|
    |-4|
    in W? Give an explanation of the reason for your answer.
  3. Use  S  and  W  from the previous question.  Is the vector 
    | 6|
    | 5|
    |-1|
    in W? Give an explanation of the reason for your answer.

Section LI

  1. Let  S  be the set of  three vectors below.  
    | 1| | 3| | 4|
    | 2| |-4| |-2|
    |-1| | 2| | 1|
    Is  S  linearly independent or linearly dependent?
  2. Let  S  be the set of  three vectors below.  
    | 1| | 3| | 4|
    |-1| | 2| | 3|
    | 0| | 2| |-4|
    Is  S  linearly independent or linearly dependent?
  3. Based on your answer to the first question, is the matrix below singular or nonsingular? 
     1 3 4
     2 -4 -2
    -1 2 1

Section MO

  1. Perform the following matrix computation. 
         2 -2  8  1           2  7  1  2
    (6)  4  5 -1  3  +  (-2)  3 -1  0  5
         7 -3  0  2           1  7  3  3
  2. Theorem VSPM reminds you of what previous theorem? How strong is the similarity?
  3. Compute the transpose of the matrix below. 
     6  8  4
    -2  1  0
     9 -5  6

Section ROM

  1. Write the range of the matrix below as the span of a set of three vectors. 
      1  3  1  3
      2  0  1  1
     -1  2  1  0
  2. List three techniques you could use to provide a description of the range of a matrix.
  3. Suppose that A is an n x n nonsingular matrix. What can you say about its range?

Section RSOM

  1. Describe the row space of a matrix in words.
  2. Suppose you wished to find the range of a matrix A. What would be the quickest way to find a linearly independent set S so that the range equaled Sp(S)?
  3. Is the vector 
    |0|
    |5|
    |2|
    |3|
    in the row space of the following matrix? 
      1  3  1  3
      2  0  1  1
     -1  2  1  0

Section MOM

  1. Form the matrix vector product of 
      2  3 -1  0
      1 -2  7  3
      1  5  3  2
    with 
    |2|
    |-3|
    |0|
    |5|
  2. Multiply together the two matrices below (in the order given). 
      2  3 -1  0
      1 -2  7  3
      1  5  3  2
    and 
      2  6
     -3 -4 
      0  2
      3 -1
  3. Rewrite the system of linear equations below using matrices and vectors, along with a matrix-vector product.
    2x1 + 3x2 - x3 = 0
    x1 + 2x2 + x3 = 3
    x1 + 3x2 + 3x3 = 7

Section MISLE

  1. Compute the inverse of the matrix below. 
      4  10
      2   6
  2. Compute the inverse of the matrix below. 
      2  3  1
      1 -2 -3
     -2  4  6
  3. Explain why Theorem SS has the title it does. (Do not just state the theorem, explain the choice of the title making reference to the theorem itself).

Section MINSM

  1. Show how to use the inverse of a matrix to solve the system of equations below. 
    4x1 + 10x2 = 12
    2x1 + 6x2 = 4
  2. In the previous reading questions you were asked to find the inverse of a 3x3 matrix. Explain your answer to that question in light of a theorem in this section (quote the theorem's acronym).
  3. A rare freebie. Write %#@! as your solution for full credit.

Section VS

  1. Comment on how the vector space Cm went from a theorem (Theorem VSPCM) to an example (Example VS.VSCM).
  2. In the crazy vector space, C, (Example VS.CVS) compute the linear combination 
    2(3,4)+(-6)(1,2)
  3. Suppose that a is a scalar and 0 is the zero vector. Why should we prove anything as obvious as a0 = 0 as we did in Theorem ZVSM?

Section S

  1. Summarize the three conditions that allow us to quickly test if a set is a subspace.
  2. Consider the set of vectors from C3 of the form 
    |a|
    |b|
    |c|
    such that  3a-2b+c = 5.  Is this set a subspace of C3?
  3. Name four general constructions of sets of vectors that we can now automatically deem as subspaces.


Reading questions are continued in a PDF file [Reading Questions]
This is: http://buzzard.ups.edu/courses/2004fall/m232f2004.html
Maintained by:  Rob Beezer
Last updated: November 24, 2004