As you know, your second exam will take place this coming Wednesday, 23 October 2013, in class. No graphing calculators will be allowed. The exam will cover material from:
Section 3.1: Determinants: Calculating them, using them to solve systems of linear equations, knowing their properties including special cases in which the determinant is easy to calculate, using them to find eigenvalues and eigenvectors, using Cramer’s rule.
Section 3.2: Solving systems of linear equations by elimination: Gaussian elimination, row echelon form, three possibilities for a system: (i) a unique solution (ii) infinitely many solutions (iii) no solution. Finding LU factorization of A aka LU decomposition of A and using it to solve Ax=b. Elimination by pivoting aka Gauss-Jordan elimination.
Section 3.3: Inverse of a matrix: Definition of an inverse, how to prove one matrix is the inverse of another (use the definition), what does it mean when the matrix has an inverse, computing the inverse, properties of the inverse. Eigenvalue decomposition of A aka diagonalization of A — what is it, how do you do it, why is it useful (study carefully pp. 204-207 of the text).
Section 3.4: Iteration: Determining dominant eigenvalue and corresponding eigenvalue by iteration (Example 2 on p. 216), solving Leontief model by iteration using properties of the geometric sum (Example 4 on p. 221, and the preceding analysis). Solution by iteration (p.223). Rewriting Ax=b to be in form x = Dx + c (sometimes c = b, sometimes not — see Example 5 (which we did in class)). Theorem 3 on p. 230.
Everything I did in class that’s not in the textbook is also fair game. Specifically, you should know that theorem I presented on 9 October with a whole bunch of conditions that are equivalent. I neglected to name it when I presented it, but I will call it … the Fundamental Theorem of Invertible Matrices.
Come see me in office hours on Tuesday, and take advantage of the TAs’ hours as well.
For practice, you should review carefully all the examples I presented in class, redo the homework problems (without reference to your previous solutions, my posted solutions, or the textbook), and do similar problems in the textbook. Like these:
Section 3.1: 1, 3, 5, 7, 17, 20, 23, 27
Section 3.2: 3ef, 4ef, 5, 9c, 15 (also by pivoting), 22
Section 3.3: 10, 19, 21, 22, 25, 29, 31, 32, 33
Section 3.4: I haven’t yet decided how exactly I’m going to test your knowledge of this part, since a lot of the implementation of what’s going on here requires a computer, but study the examples closely. Exercises: 7, 8, 13b, 14
You don’t have to do all of them because that would take a while, probably. Just make sure you feel like you could solve them all.
You should check out this album I’m listening to right now, I’m not going to say the name of the band because it’s got an obscenity in it but if you can get over that fact the album is called Slow Focus, it’s tight.
How great is Jaleel White’s face in this poster?