last one! Due Wednesday, May 1 in class
please report any typos/issues. Note that problem 7 won't be graded. Minor typos fixed 4/29
You can check your scores here
(01/14/19) systems of linear equations, matrices, Gaussian elimination
(01/16/19) vector spaces, subspaces, linear maps
(01/18/19) linear independence, spanning, basis
(01/23/19) a vector space spanned by a finite set has a finite
basis, any two finite bases of a vector space are of the same
(01/25/19) A subspace of a finite dimensional vector space is
finite dimensional. A basis of a subspace can be extended to a basis
of the whole space.
(01/28/2019) Sums of subspaces. Linear maps. Range, null space
(kernel), rank and nullity. Rank/nullity theorem.
(01/30/2019) no class due to weather.
(02/01/2019) Proof of rank/nullity ("dimension") theorem. Null
space measures injectivity. Rank and nullity of a matrix. Some
consequences of the rank/nullity theorem.
(02/04/2019) Linear bijections are isomorphisms, bases define
coordinates (isomorphisms with R^n), a linear map is uniquely
determined by what it does to a basis, isomorphisms send bases to
(02/06/2019) Linear maps and matrices, composition of linear maps and matrix multiplication.
(02/08/2019) Linear maps between vector spaces with bases and matrices.
(02/11/2019) invertible matrices, change of bases (and solutions to homework 3 problem 8 on last page)
(02/13/2019) elementary row operations and elementary matrices
(02/15/2019) (reduced) row echelon form of a matrix, rank.
(02/18/2019) Space of solutions to a system of linear equations. Quotient vector spaces.
15 (02/20/2019) Dimension of a quotient vector space. First
isomorphism theorem and some of its consequences.
16 (02/22/2019) Dual vector spaces, dual maps, transpose of a matrix.
17 (02/25/2019) Double duals, coordinates in terms of dual bases, multilinear maps.
18 (02/27/2019) Alternating multilinear maps, the group of
review problems for the first midterm.
(03/04/2019) first midterm
19 (03/06/2019) Transpositions, any permutation is a (non-unique)
product of transpositions. Implications of this fact for alternating
20 (03/08/2019) sign of a permutation, construction of the determinant.
21 (03/11/2019) properties of determinants: determinant of the
transpose, determinants and row operations, determinants of upper
22 (03/13/2019) A matrix is invertible if and only if it has
nonzero determinant, cofactors,formula for an inverse of a matrix in
terms of cofactors, similarity, determinant of a linear map.
23 (03/15/2019) More on the determinant of a linear map. Trace.
Eigenvectors and eigenvalues.
24 (03/25/2019) Characteristic polynomials. Roots and
multiplicities. (Algebraic) multiplicities of eigenvalues.
25 (03/27/2019) the product of eigenvalues is determinant, the
sum of eigenvalues is trace, sufficient condition for
problems for the second midterm.
(04/01/2019) second midterm exam
26 (04/03/2019) Inner products, Cauchy-Schwarz inequality,
27 (04/05/2019) linear and anti-linear maps, complex conjugate
vector spaces, triangle inequality, orthogonality. A collection of
orthogonal vectors is linearly independent.
28 (04/08/2019) Gram-Schmidt, orthogonal projections
29 (04/10/2019) more on orthogonal projections, adjoint operators
30 (04/12/2019) Least squares problem, orthogonal projections and
adjoint matrices. Isometries.
31 (04/15/2019) Isometries. Unitary maps. Unitary and
orthogonal matrices. Determinants and eigenvalues of unitary
matrices. Unitary equivalence.
32 (04/17/2019) Schur's theorem: for any map there is an
orthonormal basis so that the corresponding matrix is upper
triangular. Spectral theorem: any self-adjoint operator is
33 (04/19/2019) Spectral theorem. Definition of Jordan normal form. Evaluating polynomials on linear maps.
34 (04/22/2019) Algebras, homomorphisms, ideals, minimal
polynomials, Cayley-Hamilton (statement), algebraic and geometric
multiplicities of eigenvalues, generalized eigenspaces.
35 (04/24/2019) Direct sum decomposition into generalized
eigenspaces. Dimension of a generalized eigenspace is the algebraic
multiplicity of the eigenvalue.
36 (04/26/2019) Direct sum decomposition into generalized
eigenspaces. Proof of Cayley-Hamilton theorem. Example of a
computation of Jordan normal form.
37 (04/29/2019) Procedure for computing the Jordan normal form
and an example of computation of Jordan normal form. Application to
ordinary differential equations.
problems (05/01/2019) Second draft. Please report any issues
to Frank Porter's Quantum mechanics page. The math is done
carefully (unlike lots of quantum mechanics books).
Last modified: Wed May 1 12:22:20 CDT 2019