Final exam group 9, Friday 12/12/03, 5-8pm.

Tues Aug 26	Matrix multiplication is associative. Converting systems of linear equations into the form Mv=0. Starting Gaussian elimination (1.1).
Thu Aug 28	Gaussian elimination, row echelon form vs. reduced row echelon form (1.2). How to recognize from this form whether we expect 0, 1, or infinitely many solutions. Standard algebraic properties of matrices (1.3), e.g. distributivity of multiplication over addition. "Cute side topic" modeling complex numbers and infinitesimals using matrices.
Tues Sep 2	Elementary matrices, invertibility. Every square matrix can be written as a product of elementary matrices, with possibly a coordinate projection in the middle. The matrix is invertible if and only if the projection step is unnecessary.
Thu Sep 4	Column operations preserve number of, but not set of, solutions to linear systems. Using downward row and rightward column operations, we can always reduce to a 0,1 matrix with no two 1s in same row or column. If we can reduce to the identity, then the matrix has an LU decomposition. (Note: our construction of this LU decomposition was different than the one in the book, but that's okay. See Sep 11 below.)
Tues Sep 9	Geometric interpretation of some 2x2 matrices: diagonal, rotation, and shear. If the columns are parallel, the matrix is not invertible. Orthonormal sets of vectors. Expanding a vector in an orthonormal set using dot products.
Thu Sep 11	The book's LU decomposition was about writing L^-1 M = U. In class we showed that in general one could use some L and U to write L^-1 M U^-1 = some 0,1 matrix with no two 1s in same row or column. Usually that right-hand-side is the identity, so L^-1 M U^-1 = identity, giving a different construction of M = LU. Dot products (from 3.1). A physical interpretation: Force dot d(distance) measures input of energy to a physical system. V dot V = length(V)^2. V dot W = |V| |W| cos(angle between). Definition of the span of (w_1..w_k). If every w_i is orthogonal to some v, then everything in the span is also orthogonal to v. Definition of transpose. The columns of W is an orthonormal set if and only if W^T W = identity.
Tues Sep 16	The circle story: either x^2+y^2=1, or by parametrizing with angles. What we do with "spans" is the parametrizing end of things. Just as it's nice to wrap up many number equations into one vector equation, it's nice to wrap up equations with many vectors into equations with one matrix.
Thurs Sep 18	No class
Tues Sep 23	Gram-Schmid. "Orthogonal matrices". Cute side topic: any invertible square matrix is the product, uniquely, of an orthogonal matrix, positive real diagonal matrix, and upper triangular with 1s on the diagonal. (We didn't prove this, but you might convince yourself that it's Gram-Schmid in disguise.)
Tues Oct 7	Subspaces, null spaces, images. Spans are subspaces, null spaces are subspaces. The intersection of two subspaces is a subspace. (It's less obvious that the intersection of two spans is a span, except by way of subspaces.)
Thurs Oct 9	Definition of linear independence, and four characterizations. Definition of basis.
Tues Oct 14	The exchange lemma. All bases are the same size. Given a linearly independent set, and a spanning set, one can extend the independent set to a basis using only vectors from the spanning set.
Thurs Oct 16	Nullity plus rank theorem.
Tues Oct 21	Linear transformations. Every linear transformation from R^n to R^k is given by a matrix.
Thurs Oct 23	Bases give correspondences between V and R^n. With this, we can think about any composition of linear transformations in terms of matrix multiplication. This can be handy (first example) even when our linear transformation is already a matrix, using a different basis.

Coming now: differential equations.