- Section A (11:00) - HTML format, without daily schedule
- Section B (12:00) - HTML format, without daily schedule
- Section A (11:00) - PDF format, complete (requires viewer or plug-in)
- Section B (12:00) - PDF format, complete (requires viewer or plug-in)

**Homework exercises**

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

1.1 | 11 | 1, 2, 8, 11, 14, 27, 31, 34, 42 | 38 |

1.2 | 24 | 3, 5, 8, 13, 15, 17, 21, 23, 27, 29, 31, 38, 47, 49, 53 | |

1.3 | 36 | 1, 3, 5, 6, 7-19 odd, 25 | |

1.5 | 57 | 1, 3, 7, 11, 15, 23, 31, 33, 34, 35, 40, 45, 55, 63 | 59, 60, 67 |

1.6 | 68 | 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 | 47, 49, 50, 51 |

1.9 | 102 | 3, 7, 19, 23, 29, 37, 39 | 52, 53, 54, 56, 66 |

2.1 | 116 | 5, 7, 13, 15, 23, 25, 28 | |

2.2 | 124 | 3, 5, 7, 15, 17 | 18, 21, 27, 30, 31, 32 |

2.3 | 137 | 15, 17, 19, 21, 25, 27-35 odd, 39, 41, 47 | 50, 51, 52 |

2.4 | 150 | 1, 3, 7, 9, 11, 13, 19, 23, 27, 33 | 30, 38 |

2.5 | 162 | 7, 8, 9, 17, 23, 25, 27, 29 | 30, 31, 32, 36, 38, 40 |

2.6 | 174 | 3, 5, 9, 12, 13 | 22, 25, 28 |

2.7 | 190 | 1ab, 2ab, 3ab, 5, 7, 11, 13, 15, 17, 19, 29 | 33, 37, 38 |

3.2 | 237 | 1-4, 7, 9, 11, 17, 18, 19 | 23, 24, 33, 34 |

3.1 | 229 | 3, 5, 7, 9, 15 | 17, 19 |

3.4 | 254 | 3, 5, 7, 9, 13, 21 | 15, 25, 30 |

3.5 | 262 | 3, 5, 7, 9, 13, 17, 19, 27 | 21, 22, 23, 24, 25, 28, 29 |

3.6 | 273 | 7, 9, 11, 15, 21, 23, 33 | 36, 37, 38, 40, 41 |

3.7 | 285 | 3, 5, 7, 15, 17, 21 | 25, 26, 27, 29, 30, 43 |

4.2 | 314 | 1, 2, 3, 5, 9, 11, 13, 15, 18, 19 | 21, 34, 36 |

4.3 | 321 | 1, 3, 5, 7, 9, 13, 17, 19, 23, 27, 32 | 28, 29, 30 |

4.4 | 334 | 1, 3, 5, 7, 13, 14, 15, 17, 19, 21, 24, 27, 31 | 32, 36, 37, 38 |

4.5 | 338 | 1, 4, 5, 7, 9, 11, 13 | 2, 17, 18 |

4.7 | 358 | 5, 7, 9, 13, 16, 17 | 18, 19, 20, 21, 22, 26 |

4.8 | 366 | 1-6, 7, 9, 11 | 18, 19, 20, 21, 23-28 |

4.9 | 377 | 1-10, 13, 14-16, 19 | 28, 30 |

4.10 | 386 | 1, 3, 6, 9, 10, 11, 15, 16 | 17, 18, 19, 20 |

**Reading Questions**

After reading each section, send me an email (beezer@ups.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 properly and graded.

- Make your subject line exactly, exactly as follows: Math 232A x.x, where x.x is the chapter and section numbers. A is for the 11:00 section and the 12:00 section should replace it by a B. So, for example, the first reading assignment answers for someone in the 12:00 section would have the subject line: Math 232B 1.1
- Put your full name as the first line of the body of your message.
- Answer the questions in order.
- Answers are due at 7:00 in the morning on 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]

[2.1][2.2][2.3][2.4][2.5][2.6][2.7]

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

[4.1/4.2][4.3][4.4][4.5][4.7][4.8][4.9][4.10]

- 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

- This question is a freebie. Write ZZXXCV as your answer to get a free point.

- 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.
- 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?
- A freebie. Write "789" as your answer to get credit.

- Without 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 4.1 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 back in Chapter 2?
- 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 decribing 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?
- A freebie. Write 5%3# to get credit.

- 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.

- Last freebie: send in &&** to get credit.
- Give the matrix represntation of the linear transformation T:P_1 -> M_12 decribed by T(a+bx) = [2a-b a+3b] using natural bases for each space.
- 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@ups.edu, Spring 2001.