Here’s what your exam will cover:
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 (eg diagonal or triangular matrices), 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.
Section 3.5: Condition number
Also, the fundamental theorem of invertible matrices, which is this
The following are equivalent (for A and n x n matrix, b an n-vector:
(i) A is invertible
(ii) Ax = b has a unique solution
(iii) Ax = b has only the trivial solution x = 0
(iv) The reduced row echelon form of A is I
(v) rank(A) = n (i.e., there is no row of all zeros in row echelon form)
(vi) det(A) is not 0
So I’ve got a whole bunch of different types of cheese in my refrigerator right now and I wanted to talk about them.
First of all, I have three different types from Sprout Creek Farm in Poughkeepsie — Toussaint (raw cow’s milk, aged 5-7 months), Smoked Toussaint (same, but smoked), and a raw goat’s milk cheese whose name escapes me. I also have some aged Gouda and Ewephoria from East Village Cheese that are each almost kicked but still have a bit left. And yet despite this bounty, when on Sunday I found myself on Bleecker Street I just had to stop into Murray’s for more cheese. And I’m glad I did because they had a great sale going. I got:
Chevre d’argental (goat, France)
Brebirousse d’argental (sheep, France)
Boerenkaas Gouda (cow, Holland)
Bleu d’auvergne (cow, France)
Making a good cheese board is all about balance and harmony. The fresh, clean chevre starts things off nice and easy, then the slightly funkier but still stupendously smooth Brebirousse (they’re from the same region in France, as you can tell by the names) gets you a little more revved up. I should’ve gotten a semi-firm cheese to follow up — maybe an Alpine cheese like the oniony Scharfe Maxx or classic Appenzeller — but I demurred. I always like to end with a well-aged, unpasteurized, tyrosine-laden cheese and a good blue, so the latter two fit the bill. All but the Brebirrouse were on sale, so I was pretty stoked; usually when I get as much cheese as I did on Sunday I’m out like $50. Plus they had Tom Cat baguettes (best in the city besides Pain d’Avignon, but much more widely available) on 2-for-1 sale, so that was dope.
Ah! I got some fresh ricotta, too. Been eating it on dem Tom Cats with Bonne Maman strawberry jam. DAAAAAAAAMN!
Prof. Prattle 