## Spring 2019, section C3H

• Assignment #1 Due Wednesday, January 23 in class
Read Sections 1.1 -- 1.3 of Treil's Linear algebra done wrong
Exercises

• Assignment #2 Due Wednesday, January 30 Friday, February 1 in class
Exercises

• Assignment #3 Due Wednesday, February 6 in class
Exercises

• Assignment #4 Due Wednesday, February 13 in class
Exercises Note that problems 3 and 4 are optional.

• Assignment #5 Due Wednesday, February 20 in class
Exercises (dyslexic mistakes in problems 1, 2 fixed 2/17)).

• Assignment #6 Due Wednesday, February 27 in class
Exercises

• Assignment #7 Due Wednesday, March 6 in class
Exercises Solutions

• Assignment #8 Due Wednesday, March 13 in class
Exercises

• Assignment #9 Due Wednesday, March 27 in class
Exercises

• Assignment #10 Due Wednesday, April 3 in class
Exercises

• Assignment #11 Due Wednesday, April 10 in class
Exercises (Signs in problem 6 fixed on 4/7)

• Assignment #12 Due Wednesday, April 17 in class
Exercises

• Assignment #13 Due Wednesday, April 24 in class
Exercises please report any typos/issues. (A typo in Problem 1 fixed on 4/21, more small changes 4/22)

• Assignment #14 last one! Due Wednesday, May 1 in class
Exercises please report any typos/issues. Note that problem 7 won't be graded. Minor typos fixed 4/29

You can check your scores here

lecture 1 (01/14/19) systems of linear equations, matrices, Gaussian elimination
lecture 2 (01/16/19) vector spaces, subspaces, linear maps
lecture 3 (01/18/19) linear independence, spanning, basis
lecture 4 (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 size.
lecture 5 (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.
lecture 6 (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.
lecture 7 (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.
lecture 8 (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 bases.
lecture 9 (02/06/2019) Linear maps and matrices, composition of linear maps and matrix multiplication.
lecture 10 (02/08/2019) Linear maps between vector spaces with bases and matrices.
lecture 11 (02/11/2019) invertible matrices, change of bases (and solutions to homework 3 problem 8 on last page)
lecture 12 (02/13/2019) elementary row operations and elementary matrices
lecture 13 (02/15/2019) (reduced) row echelon form of a matrix, rank.
lecture 14 (02/18/2019) Space of solutions to a system of linear equations. Quotient vector spaces.
lecture 15 (02/20/2019) Dimension of a quotient vector space. First isomorphism theorem and some of its consequences.
lecture 16 (02/22/2019) Dual vector spaces, dual maps, transpose of a matrix.
lecture 17 (02/25/2019) Double duals, coordinates in terms of dual bases, multilinear maps.
lecture 18 (02/27/2019) Alternating multilinear maps, the group of permutations.
review problems for the first midterm.
(03/04/2019) first midterm
lecture 19 (03/06/2019) Transpositions, any permutation is a (non-unique) product of transpositions. Implications of this fact for alternating n-linear maps.
lecture 20 (03/08/2019) sign of a permutation, construction of the determinant.
lecture 21 (03/11/2019) properties of determinants: determinant of the transpose, determinants and row operations, determinants of upper triangular matrices...
lecture 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.
lecture 23 (03/15/2019) More on the determinant of a linear map. Trace. Eigenvectors and eigenvalues.
lecture 24 (03/25/2019) Characteristic polynomials. Roots and multiplicities. (Algebraic) multiplicities of eigenvalues.
lecture 25 (03/27/2019) the product of eigenvalues is determinant, the sum of eigenvalues is trace, sufficient condition for diagonalizability.
review problems for the second midterm.
(04/01/2019) second midterm exam
lecture 26 (04/03/2019) Inner products, Cauchy-Schwarz inequality,
lecture 27 (04/05/2019) linear and anti-linear maps, complex conjugate vector spaces, triangle inequality, orthogonality. A collection of orthogonal vectors is linearly independent.
lecture 28 (04/08/2019) Gram-Schmidt, orthogonal projections
lecture 29 (04/10/2019) more on orthogonal projections, adjoint operators and matrices.
lecture 30 (04/12/2019) Least squares problem, orthogonal projections and adjoint matrices. Isometries.
lecture 31 (04/15/2019) Isometries. Unitary maps. Unitary and orthogonal matrices. Determinants and eigenvalues of unitary matrices. Unitary equivalence.
lecture 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 diagonalizable.
lecture 33 (04/19/2019) Spectral theorem. Definition of Jordan normal form. Evaluating polynomials on linear maps.
lecture 34 (04/22/2019) Algebras, homomorphisms, ideals, minimal polynomials, Cayley-Hamilton (statement), algebraic and geometric multiplicities of eigenvalues, generalized eigenspaces.
lecture 35 (04/24/2019) Direct sum decomposition into generalized eigenspaces. Dimension of a generalized eigenspace is the algebraic multiplicity of the eigenvalue.
lecture 36 (04/26/2019) Direct sum decomposition into generalized eigenspaces. Proof of Cayley-Hamilton theorem. Example of a computation of Jordan normal form.
lecture 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.
review problems (05/01/2019) Second draft. Please report any issues

A link to Frank Porter's Quantum mechanics page. The math is done carefully (unlike lots of quantum mechanics books).