# Math 304, Section 100, Linear Algebra, Summer 2008

Harold P.
Boas

- Saturday, June 28
- The second exam scores and the final course grades are posted at the TAMU online learning site.

Congratulations to the 38% of the class that made an A on the second exam, and kudos to the three students who scored 100. The median score on the exam was 78, and the low score was 42.

Some students blanked out on the second exam, so I gave everybody a boost in the course average by using a weighted norm. Instead of basing the letter grade on the average of the three scores (the quiz average, Exam 1, and Exam 2), I used the average of four numbers, the fourth one being the maximum of the three scores. In other words, I counted the maximum of the three scores twice. This algorithm can be expressed as follows in the language of norms that we learned. If**v**denotes the vector of three scores, then the usual average is (1/3)||**v**||_{1}. The weighted average is (1/4)(||**v**||_{1}+||**v**||_{∞}).

The class mean for the weighted course average was 86. - Friday, June 27
- The second (and final) exam was given, and solutions are available.
- Thursday, June 26
- We reviewed for the examination to be given tomorrow.
- Wednesday, June 25
- We discussed diagonalization of matrices, the exponential of a matrix, and the solution of systems of differential equations by using the exponential matrix. The sixteenth quiz was given, and solutions are available. The slides from class are available too.
- Tuesday, June 24
- We discussed the application of eigenvalues and eigenvectors to the solution of linear systems of differential equations. The slides from class are available. We worked in groups on exercise 2(d) from page 323 for a quiz grade.
- Monday, June 23
- We discussed eigenvalues and eigenvectors. The slides from class are available. We worked in groups on problem 10 from the Chapter 5 Test B for a quiz grade.
- Friday, June 20
- We discussed the Gram-Schmidt orthogonalization process and saw how interpreting the process as multiplying a matrix by a succession of elementary matrices leads to the QR factorization. The thirteenth quiz was given, and solutions are available.
- Thursday, June 19
- We discussed the notions of orthogonal sets of vectors, orthonormal sets of vectors, trigonometric polynomials, Fourier coefficients, and orthogonal matrices. We worked in groups on problem 10 from the Chapter 3 Test B for a quiz grade.
- Wednesday, June 18
- We discussed the notions of inner product (a generalization of the ordinary scalar product of vectors) and norm (a generalization of the ordinary length of a vector). The eleventh quiz was given, and solutions are available.
- Tuesday, June 17
- We discussed how to find an optimal approximate solution to an inconsistent linear system by the method of least squares: namely, replace the equation Ax=b with the equation A
^{T}Ax=A^{T}b. Then we worked in groups on problems 2, 4, and 6 from Chapter Test B of Chapter 4 for a quiz grade. - Monday, June 16
- We discussed orthogonal subspaces: in particular, the equality between the null space of a matrix and the orthogonal complement of the range of the transpose. The ninth quiz was given, and solutions are available.
- Friday, June 13
- We discussed the notions of scalar product, orthogonality, projection, and the Cauchy-Schwarz inequality.
- Thursday, June 12
- We discussed how the matrix representation of a linear transformation depends on the basis of the vector space, and we saw that this study leads to the notion of similar matrices. The eighth quiz was given, and solutions are available.
- Wednesday, June 11
- We discussed the representation of linear transformations by matrices. We worked in groups on Chapter Test A from Chapter 3 for a quiz grade.
- Tuesday, June 10
- We discussed the concepts of linear transformations, linear operators, the kernel of a linear transformation, and the image of a subspace under a linear transformation. The sixth quiz was given, and solutions are available.
- Monday, June 9
- We discussed the notions of row space, column space, rank of a matrix, and nullity (dimension of the null space). We saw why the row space and the column space have the same dimension (both dimensions equal the number of leading 1's in the reduced row echelon form), and we saw why the sum of the rank and the nullity equals the number of columns of the matrix (the rank equals the number of leading 1's in the reduced row echelon form, and the nullity equals the number of free variables).
- Saturday, June 7
- The class did a good job on the first exam. The median score was 85. I will hand back the graded exams in class on Monday. In the meantime, you can check your score online at the TAMU eLearning site.
- Friday, June 6
- The first exam was given, and solutions are available.
- Thursday, June 5
- The Mathematics Department now has a help session for Math 304 on
Tuesdays and Thursdays, 14:00–16:30, in Blocker 624.

In class, we reviewed for the exam to be given tomorrow on sections 1.1–1.4, 2.1, 2.2, and 3.1–3.4. We have talked about section 3.5 on change of basis, but section 3.5 will not be covered on this exam.

You may wish to look at my exam from 2007 and the solutions as well as my exam and solutions from 2006. - Wednesday, June 4
- We discussed the notions of basis, dimension, and change of basis. The fifth quiz was given, and solutions are available.
- Tuesday, June 3
- We discussed the notions of linear dependence and linear independence. We worked in groups on the true/false Chapter Test A for Chapter 1, which we rolled over into a group take-home quiz due tomorrow.
- Monday, June 2
- We discussed the notions of vector spaces, subspaces, null spaces, span, and spanning sets. The third quiz was given, and solutions are available.
- Friday, May 30
- We discussed determinants and their properties. The first two quizzes were returned. You can check your grades for this course online at the TAMU eLearning site.
- Thursday, May 29
- We discussed the so-called elementary matrices, the computation of inverse matrices, and the LU factorization. The second quiz was given, and solutions are available.
- Wednesday, May 28
- We discussed algebraic operations on matrices, the interpretation of a linear system of equations as a matrix equation, and the Consistency Theorem.

The first quiz was given, and solutions are available. - Tuesday, May 27
- We discussed the solution of systems of linear equations by the method of row reduction. Here is a vocabulary list for sections 1.1 and 1.2:
- equivalent linear systems
- consistent/inconsistent linear systems
- overdetermined/underdetermined linear systems
- augmented matrix of a linear system
- homogeneous linear system
- elementary row operations
- (reduced) row echelon form
- Gaussian elimination

- Monday, May 26
- I posted the syllabus for the course.
- Thursday, May 22
- 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.