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).

- Thinking Mathematically, Mason/Burton/Stacey
- How to Read and Do Proofs, Solow
- How to Prove It: A Structured Approach, Velleman
- Mathematical Thinking: Problem-Solving and Proofs, West/D'Angelo
- Proofs and Fundamentals: A First Course in Abstract Mathematics, Bloch

- Texas Instrument TI-86 Guidebook [Chapter 13]
- Math Forum Linear Algebra Pages
- A 2-D Eigenvalue Demonstration
- Echelon Form

**Homework exercises**

Section | Page | Computational | Theoretical |
---|---|---|---|

1.1 | 12 | 1, 2, 8, 11, 14, 27, 31, 34, 42 | 39 |

1.2 | 26 | 3, 5, 8, 13, 15, 17, 21, 23, 27, 29, 31, 38, 49, 50, 53 | |

1.3 | 37 | 1, 3, 5, 6, 7-17 odd, 19, 21, 25 | |

1.5 | 58 | 1, 3, 7, 11, 15, 23, 31, 33, 34, 35, 40, 45, 55, 63 | 59, 60, 67 |

1.6 | 69 | 1, 3, 5, 17, 21, 26, 27, 30, 31, 32 | 44, 46, 47 |

1.7 | 78 | 1-13 odd, 17, 23, 27, 30, 41, 43 | 49, 51, 52, 53 |

1.9 | 104 | 3, 7, 19, 23, 31, 39, 41 | 54, 55, 56, 58, 68 |

3.1 | 167 | 5, 7, 13, 15, 23, 25, 28 | |

3.2 | 175 | 3, 5, 7, 15, 17 | 18, 21, 27, 30, 31, 32 |

3.3 | 186 | 15, 17, 19, 21, 25, 27-35 odd, 39, 41, 47 | 50, 51, 52 |

3.4 | 200 | 1, 3, 7, 9, 11, 13, 19, 23, 27, 33 | 30, 38 |

3.5 | 212 | 7, 8, 9, 17, 23, 25, 27, 29 | 30, 31, 32, 36, 38, 40 |

3.6 | 224 | 3, 5, 9, 12, 13 | 22, 25, 28 |

3.7 | 239 | 1ab, 2ab, 3ab, 5, 7, 11, 13, 15, 17, 19, 29 | 33, 37, 38 |

4.2 | 288 | 1-4, 7, 9, 11, 17, 18, 19 | 23, 24, 33, 34 |

4.1 | 279 | 3, 5, 7, 9, 15 | 17, 19 |

4.4 | 305 | 3, 5, 7, 9, 13, 21 | 15, 25, 30 |

4.5 | 314 | 3, 5, 7, 9, 13, 17, 19, 27 | 21, 22, 23, 24, 25, 28, 29 |

4.6 | 324 | 7, 9, 11, 15, 21, 23, 33 | 36, 37, 38, 40, 41 |

4.7 | 336 | 3, 5, 7, 15, 17, 21 | 25, 26, 27, 29, 30, 43 |

5.2 | 368 | 1, 2, 3, 5, 9, 11, 13, 15, 18, 19 | 21, 34, 36 |

5.3 | 373 | 1, 3, 5, 7, 9, 13, 17, 19, 23, 27, 32 | 28, 29, 30 |

5.4 | 386 | 1, 3, 5, 7, 13, 14, 15, 17, 19, 21, 24, 27, 31 | 32, 36, 37, 38 |

5.5 | 390 | 1, 4, 5, 7, 9, 11, 13 | 2, 17, 18 |

5.7 | 410 | 5, 7, 9, 13, 16, 17 | 18, 19, 20, 21, 22, 26 |

5.8 | 418 | 1-6, 7, 9, 11 | 18, 19, 20, 21, 23-28 |

5.9 | 429 | 1-10, 13, 14-16, 19 | 28, 30 |

5.10 | 438 | 1, 3, 6, 9, 10, 11, 15, 16 | 17, 18, 19, 20 |

**Reading Questions**

After reading each section, send me an email (BEEZER(at)UPS(dot)EDU) 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.

- Make your subject line exactly,
exactly, exactly as follows: Math 232 x.x, where x.x is the chapter and
section numbers. So, for example, the first reading assignment answers
would have the subject line (exactly):
Math 232 1.1

Do not send me other messages about the course that begin with "Math 232" or they will get filtered, and I will not see your message until the next time I review the answers to the reading questions. - Put your full name as the first line of the body of your message.
- Answer the questions in order.
- 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**

[1.1][1.2][1.3][1.5]

[1.6][1.7][1.9]

[3.1][3.2][3.3][3.4]
[3.5][3.6][3.7]

[4.2][4.1][4.4]
[4.5][4.6][4.7]

[5.1/5.2][5.3][5.4]

[5.5][5.7][5.8]
[5.9][5.10]

- Write the augmented matrix for the system of equations: 3x - y = -1, x + 2y = 9.
- Find all solutions to the system in question 1.
- Describe the geometric picture that you would associate with this system and its solution.

Consider the matrix:

1 5 0 2 1 0 0 1 3 -2 0 0 0 0 0

- Is the matrix in reduced echelon form? Why or why not?
- Suppose this is the augmented matrix of a system of equations. Is this system inconsistent? Why or why not?
- Suppose this is the augmented matrix of a system of equations. Find a solution.

- What are the three possibilities for solutions to a system of equations?
- What is the definition of a homogenous system of equations?
- Is a homogenous system of equations always consistent? Why or why not?

- What is a vector?
- Compute the product, AB, of the two 2 x 2 matrices, A and B below.
1 2 A = 3 1

-1 4 B = 3 2

- Express the system of equations, 3x - y = -1, x + 2y = 9, using matrix multiplication.

- Define informally the transpose of a matrix.
- What interesting properties does the n x n identity matrix, I_n, have?
- Compute ||(3, -2, 1)^T||.

- Describe what it means for a set of vectors to be linearly independent.
- Define when a matrix is nonsingular.
- Is the matrix C below nonsingular?
2 4 C = 1 2

- Describe informally in words what it means for one matrix to be the inverse of another.
- Why would we care to learn about inverses of matrices?
- What keys do you use on your calculator to compute the inverse of a matrix?

- Describe scalar multiplication of a vector in terms of the geometry.
- Describe vector addition of two vectors in terms of the geometry.
- Suppose we have two vectors that lie on the same line through the origin. If we add these vectors, what can you say about the result?

- Without simply listing the definition, answer the question, "What is a subspace?"
- What do subspaces of R
^{2}look like? - Why is the word "closure" an apt choice for properties C1 and C2 of Theorem 1?

Suppose that A is a matrix. Decribe, informally, the ...

- ...null space of A.
- ...range of A.
- ...column space of A.

- What is a basis?
- Describe the "tension" in the definition of a basis.
- Suppose that
**v**is a non-zero vector from R^{2}. Could {**v**} be a basis of R^{2}?

- Why is Theorem 8 proved prior to Definition 5?
- What is the rank of a matrix?
- Suppose we have 4 vectors from R^3. How would a dimension argument establish that they are linearly dependent?

- Is the set {(1,-1,2), (5,3,-1), (8,4,-2)} an orthogonal set?
- What is the disinction between an orthogonal set and an orthonormal set?
- What is nice about the output of the Gram-Schmidt process?

Suppose that G: R^{3} -> R^{2} is a linear transformation with
G( (1, 3,2)^{T} ) = (2, 8)^{T} and
G( (5, -6, 7)^{T} ) = (1, 3)^{T}.

- What is G( (10, 30, 20)
^{T}) ? - What is G( (6, -3, 9)
^{T}) ? - Where in this section is there another place where we could use the term "representation"?

Just read about determinants and ignore material about eigenvalues.

- What is the determinant of the following matrix?
2 7 3 5

- What is the determinant of the following matrix?
-1 5 4 2 9 11 3 -1 2

- What key property of the matrix in (2) is revealed by the value of the determinant?

Consider the following 2x2 matrix:

5 -3 6 -4

- Find an eigenvector for the eigenvalue 2.
- Find another eigenvalue of this matrix.
- Report on how your calculator can be used to find eigenvalues.

- What is the definition of the characteristic polynomial?
- How is the characteristic polynomial used to find eigenvalues?
- Why can't an n x n matrix have more than n different eigenvalues?

- What is an eigenspace?
- How does geometric multiplicity differ from algebraic multiplicity?
- When is a matrix defective?

- What is (3 + 4i)
^{-1}? - What can be said about complex numbers that arise as roots of a polynomial with real coefficients?
- What is so amazing about the eigenvalues of a real symmetric matrix?

- When are two matrices similar?
- What does it mean for a matrix to be diagonalizable?
- When is a matrix orthogonal?

- Describe briefly a problem in mathematics from Section 5.1 that is helped by vector space concepts.
- How many axioms (properties) must be satisfied before a set with two operations can be called a vector space?
- Briefly give one example of a vector space that is not R
^{n}.

- What is the definition of a subspace?
- How does the newest definition of a spanning set differ from the one we saw in Chapter 3?
- When is a matrix skew-symmetric?

- Express the following matrix as a coordinate vector relative to a
natural basis:
5 -3 6 -4

- Where would it be appropriate to use the word "representation" when describing the contents of this section?
- Comment on Theorem 5.

- Why doesn't the text prove Theorem 6?
- What is the new definition of dimension?
- Where have you seen Theorems 8 and 9 before?

- What does "it is enough to know what a linear transformation does to a basis" mean?
- Does Theorem 15, part (3) look familiar? Why?
- According to theorems in this section,
linear independence, one-to-one-ness and trivial null spaces are all related.
What
*single*common word might be used in the definition of each of these three related ideas?

- What is the definition of an invertible linear transformation?
- The vector space of polynomials of degree 3 or less, P
_{3}, is isomorphic to which popular vector space? - What is the "punch-line" in this section? A punch-line is a revelation, big theorem, unifying theorem - anything that seems like a major event in the course.

- Give the matrix representation of the linear transformation T:P
_{1}-> M_{12}decribed by T(a+bx) = [2a-b a+3b] using natural bases for each space. - If A is a matrix representation of the linear transformation T relative to bases D and E, and B is the matrix representation of the linear transformation S relative to bases D and E, then how would you obtain the matrix representation of the linear transformation T+S relative to bases D and E?
- What is the "punch-line" in this section?

- What is the definition of an eigenvector of a linear transformation?
- What is a transistion matrix?
- What is the "punch-line" in this section?

Rob Beezer, BEEZER(at)UPS(dot)EDU, Spring 2003.