# Math 323–502, Linear Algebra, Fall 2008

Harold P.
Boas

- Wednesday, December 10
- The final exam was given, and solutions are available.
- Tuesday, December 2 (the last class meeting)
- We reviewed for the final examination to be given on Wednesday, December 10, at 8:00am.
- Tuesday, November 25
- We discussed the application of eigenvalues and eigenvectors to solving linear systems of differential equations. The slides from class are available.

There is no assignment to hand in next time, but some suggested exercises from §6.2 on pages 323–324 are 1a, 2b, and 6. Next Tuesday, December 2, is our last class meeting. - Thursday, November 20
- We discussed eigenvalues and eigenvectors and the diagonalization of matrices. The slides from class are available.

Here is the assignment for next time.- §6.1, pages 310–312, exercises 1(h), 6, 9, and 18.
- §6.3, page 340, exercise 1(d).

- Tuesday, November 18
- We discussed the Gram–Schmidt orthonormalization procedure and the associated QR factorization of matrices.

The assignment for next time is from §5.6, pages 281–282, exercises 3, 4, and 7. - Thursday, November 13
- We discussed orthogonal sets, orthonormal sets, Fourier coefficients, trigonometric polynomials, and orthogonal matrices. The assignment for next time is from §5.5, pages 270–273, exercises 2, 3, 6, 21, and 28.
- Tuesday, November 11
- We discussed inner product spaces and normed spaces.

The assignment for next time is from §5.4, pages 252–254, exercises 3, 7c, 15b, and 26. - Thursday, November 6
- We discussed the method of least squares for finding approximate solutions to inconsistent linear systems.

Here is the assignment for next time.- §5.2, pages 233–234, exercise 12.
- §5.3, pages 243–244, exercises 1(b), 5, 6, and 10.

- Tuesday, November 4
- We discussed orthogonal subspaces and in particular the theorem that the null space of a matrix is the orthogonal complement of the range of the transpose matrix.

The assignment for next time is from §5.2, pages 233–234, exercises 4, 6, 9, and 15. - Thursday, October 30
- The graded exams were returned. Congratulations to the quarter of the class that made A's and especially to the one person who scored 100.

For those of you who did not perform up to your potential on the second exam, here is an opportunity to earn some extra points. If your score on the final exam is better than your score on Exam 2, then I will replace your Exam 2 score by its average with the final exam score. For example, if you scored 60 on Exam 2 and you score 100 on the final exam, then I will replace that 60 with 80. Then I will compute the course grade as announced at the beginning of the semester by averaging Exam 1, Exam 2, the final exam, and the homework average.

In class we discussed the notion of scalar product in R^{2}and R^{3}, orthogonality, the Cauchy-Schwarz inequality, and projections. The assignment for next time is from §5.1, pages 223–224, exercises 1c, 2c, 5, 9, and 13. - Tuesday, October 28
- The second exam was given, and solutions are available.
- Thursday, October 23
- We reviewed for the examination to be held on Tuesday, October 28 on Chapters 3 and 4 of the textbook. The main topics are subspaces, the null space of a matrix, the span of a set of vectors, linear independence, the notions of basis and dimension of a vector space, change of basis, row space, column space, rank, nullity, the notion of a linear transformation, kernel, image, matrix representations of linear transformations, and similar matrices.
- Tuesday, October 21
- We discussed the representation of linear transformations by matrices and the notion of similar matrices. The puzzle from class is available.

Reminder: the second exam, on Chapters 3 and 4, is next Tuesday. Here are some suggested exercises for next time (not to hand in).- Section 4.2, pages 196–197, exercises 2a, 5c, and 6.
- Section 4.3, pages 205–206, exercises 4 and 5.

- Thursday, October 16
- We discussed the notions of linear transformation, linear operator, kernel, image, and range.

Here is the assignment for next time.- Section 3.6, page 169, exercise 12.
- Section 4.1, pages 182–184, exercises 4, 5, 8, and 16.

- Tuesday, October 14
- We discussed the notions of row space, column space, rank, nullity, and the rank-nullity theorem.

Here is the assignment for next time.- Section 3.5, page 161, exercise 8.
- Section 3.6, pages 167–168, exercises 1c, 8, and 10.

- Thursday, October 9
- We discussed the topic of change of basis in a vector space.

Here is the assignment for next time.- Section 3.3, page 145, exercise 11.
- Section 3.4, page 151, exercises 8 and 15.
- Section 3.5, page 161, exercises 5, 6, and 9.

- Tuesday, October 7
- We discussed the concepts of basis and dimension for vector spaces.

Here is the assignment for next time.- Section 3.3, page 145, exercise 15.
- Section 3.4, pages 150–151, exercises 3, 5, and 14ac.

- Thursday, October 2
- We discussed the concept of linear (in)dependence and looked at some ways to detect it. In particular, we saw that questions about linear independence can sometimes be answered by computing suitable determinants.

Here is the assignment for next time.- Section 3.2, page 133, exercise 14.
- Section 3.3, pages 144–145, exercises 2ac, 5, 7d, 14.

- Tuesday, September 30
- We discussed two ways that subspaces typically appear: as the null space of a matrix, and as the span of a set of vectors.

The assignment for next time is exercises 4(c), 6, and 8 on page 132 in section 3.2. - Monday, September 29
- Solutions to the first exam are available.
- Thursday, September 25
- The first exam was given.
- Tuesday, September 23
- We reviewed for the exam to be given next time.
- Thursday, September 18
- We discussed the notion of a vector space.

The assignment is to prepare for the examination to be given on Thursday, September 25. Can you solve the problems in the Chapter Tests at the end of Chapters 1 and 2? - Tuesday, September 16
- We discussed the formula for the inverse of a matrix along with Cramer's rule for solving a linear system.

Here is the assignment to hand in next time: exercises 6 and 12 on pages 104–105 in section 2.2 and exercises 3 and 7 on page 110 in section 2.3. - Thursday, September 11
- We discussed the properties of determinants, methods for computing determinants, and the
*LU*factorization of a matrix.

The assignment to hand in next time is exercises 6 and 11 on pages 97–98 in section 2.1 and exercises 4 and 11 on page 104 in section 2.2. - Tuesday, September 9
- By way of review for the first exam (to be given on September 25), we worked in groups on Chapter Test A for Chapter 1 (top of page 88). Here is the assignment to hand in next time: problems 1, 3, and 5 in Chapter Test B on pages 88–89.
- Thursday, September 4
- We discussed the computation of inverse matrices from the point of view of elementary matrices, and we saw why a square linear system has a unique solution if and only if the coefficient matrix is invertible.

Here is the assignment.- Computational exercises, not to hand in: exercises 3, 9, and 11 on pages 69–70 in section 1.4.
- Exercises requiring explanation, to hand in on Tuesday: exercises 6, 12(b), and 16 on pages 69–71 in section 1.4. (Note that the author's notation
**a**_{1}means the first column of the matrix*A*.)

- Wednesday, September 3
- The Department of Mathematics has announced the schedule for help sessions. The help session for Math 323 meets in ENPH 213 on Monday, Tuesday, Wednesday, and Thursday from 6:00pm to 8:00pm (except on 9/22, 10/13, 11/3, and 12/1, when the hours are 5:30pm to 7:30pm).
- Tuesday, September 2
- We continued the discussion of matrix algebra, including the non-commutativity of matrix multiplication, the identity matrix, inverse matrices, the property that (AB)
^{-1}=B^{-1}A^{-1}, and elementary matrices.

Here is the assignment.- Computational exercises, not to hand in: in section 1.3, do exercises 2, 4, and 10 on pages 57–59, and check your answers in the back of the book.
- Exercises requiring explanation, to hand in on Thursday: exercises 13, 17, and 23 on pages 59–60 in section 1.3.

- Friday, August 29
- In class yesterday, we talked about various terminology, including the notions of equivalent linear systems, consistent or inconsistent linear systems, overdetermined or underdetermined linear systems, homogeneous linear systems, the augmented matrix of a linear system, elementary row operations, row echelon form, and reduced row echelon form. We discussed matrix notation, the transpose of a matrix, the concept of a symmetric matrix, and rewriting a linear system in matrix form. We saw that focusing on
*columns*leads to the Consistency Theorem on page 37 of the textbook.

Here is the assignment:- Not to hand in: in section 1.2, do exercises 1, 2(a,d), and 6(a,b) on pages 25–26 and check your answers in the back of the book.
- To hand in on Tuesday: exercises 7, 8, and 9 on page 27 in section 1.2. Notice that exercises 8 and 9 have answers in the back of the book, so your task is to supply supporting explanation in complete sentences.

- Tuesday, August 26
- We looked at examples of solving a linear system of equations by converting the system into a simpler, equivalent system in triangular form. We saw that a linear system has either a unique solution, no solution, or infinitely many solutions, and we discussed the geometric interpretation of these three cases. Here is the first assignment (from section 1.1 in the textbook):
- Not to hand in: do exercises 1(b) and 6(b,d,h) on pages 11–12 and check your answers in the back of the book.
- To hand in on Thursday: exercises 9 and 10 on pages 12–13 in the textbook. Your solutions should be written in complete sentences!

- Monday, August 25
- This site went live today. The first-day handout is available online and also in a printable format.

These pages are copyright © 2008 by Harold P. Boas. All rights
reserved.