lecture 1 (08/26/24) systems of linear equations, matrices, Gaussian elimination. Corresponding sections in Friedberg/Insel/Spence (FIS) --- pp. 26-29 and Sec. 3.4
lecture 2 (08/28/24) Definition of a (real) vector space, elementary properties, subspaces (corresponding sections in FIS --- Sections 1.2, 1.3 and 2.1)
lecture 3 (08/30/24) Definition of a linear map. linear independence, spanning, basis
(FIS --- Sections 1.5, 1.6, and 2.1)
lecture 4 (09/04/24) 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 (09/06/24) A subspace of a finite dimensional vector space is finite dimensional and related facts.
lecture 6 (09/09/24) Sums of subspaces. Properties of linear maps. Range and null subspaces associated to a linear map. Rank and nullity.
lecture 7 (09/11/24) Rank-nullity theorem and applications. Null spaces measure injectivity. Applications to solutions of linear equations.
lecture 8 (09/13/24) Linear bijections have linear inverses. Bases define coordinates. A linear map is uniquely determined by what it does to a basis.
lecture 9 (09/16/24) Isomorphisms send bases to bases. Linear maps and matrices, composition of linear maps and matrix multiplication.
lecture 10 (09/18/24) Matrices for linear maps between vector spaces with ordered bases. Invertible matrices and their uniqueness.
lecture 11 (09/20/24) Matrix for a composition of two linear maps. Change of bases.
lecture 12 (09/23/24) Elementary row operations and elementary matrices. Computation of the inverse of a matrix. Definition of a row echelon form.
lecture 13 (09/25/24) Reduced row echelon form; rank and nullity in terms of pivots.
lecture 14 (09/27/24) Space of solutions of a system of linear equations. Quotient of a vector space by a subspace.
Additional resources: Wikipedia page on quotient vector spaces . Notes on quotient vector spaces by Santiago Canez.
lecture 15 (09/30/24) Dimension of the quotient vector space. First and second isomorphism theorems.
lecture 16 (10/02/24) Rank/nullity from the 1ist isomorphism theorem. Second isomorphism theorem. Dual vector spaces. Transpose of a matrix.
review for the first midterm (10/04/24) Please report any issues on Campuswire.
first midterm (10/07/24)
lecture 17 (10/09/24) Dual basis. Transpose of a product of matrices. Multilinear maps.
lecture 18 (10/11/24) Definition of a group. The permutation group Sn and alternating multilinear maps. Transpositions.
lecture 19 (10/14/24) Any permutation is a product of transpositions. A definition of determinant.
lecture 20 (10/16/24) Construction of the sign homomorphism. Determinant is alternating.
lecture 21 (10/18/24) Properties of determinant. In particular determinants of upper triangular matrices, det(AT) = det(A), determinants and row operations and det(AB) = det(A)det(B) for all A,B. A matrix is invertible iff its determinant is nonzero.
lecture 22 (10/21/24) Expansion of a determinant along a column and a row. Cofactors, formula for the inverse of a matrix in terms of cofactors.
lecture 23 (10/23/24) Conjugate/similar matrices. Trace of a matrix and of a linear map. Eigenvectors and eigenvalues.
lecture 24 (10/25/24) Characteristic polynomials. Roots and multiplicities. (Algebraic) multiplicities of eigenvalues.
The product of eigenvalues is determinant, the sum of eigenvalues is trace.
lecture 25 (10/28/24) Diagonalizability of linear maps and of matrices and why one may care. A map with all eigenvalues distinct is diagonalizable.
lecture 26 (10/30/24) Hermitian vector spaces. Norms. Cauchy-Schwarz and triangle inequalities.
review for the second midterm (11/01/24) Please report any issues on Campuswire.
second midterm (11/04/24)
lecture 27 (11/06/24) Orthogonal and orthonormal bases. Gram-Schmidt. Projections.
lecture 28 (11/08/24) Complex conjugate vector spaces. Projections. Orthogonal projections.
lecture 29 (11/11/24) Orthogonal projections. A vector in a subspace closest to a given vector. Adjoint linear maps and matrices.
lecture 30 (11/13/24) Conjugate transpose matrices and adjoint maps. Least squares problem, related problems and orthogonal projections.
lecture 31 (11/15/24) Isometries. Unitary maps. Unitary and orthogonal matrices. Determinants and eigenvalues of unitary matrices. Unitary equivalence.
lecture 32 (11/18/24) 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 in an orthonormal basis.
lecture 33 (11/20/24) Spectral theorem for matrices. Definition of Jordan normal form. Plugging linear maps (and matrices) into polynomials.
lecture 34 (11/21/24) Algebras, homomorphisms(of algebras), ideals, minimal polynomials, Cayley-Hamilton (statement), algebraic and geometric multiplicities of eigenvalues.
lecture 35 (12/02/24) Direct sum decomposition into generalized eigenspaces. Dimension of a generalized eigenspace is the algebraic multiplicity of the eigenvalue
lecture 36 (12/04/24) Jordan normal form and Cayley-Hamilton for nilpotent linear maps.
lecture 37 (12/06/24) Jordan normal form and Cayley-Hamilton for arbitrary linear maps.
lecture 38 (12/09/24) "Abstract" Jordan normal form. Exponentials of matrices. Linear ODEs.
review (12/09/24)
(The second half of the lectures has videos)
lecture 1 (01/22/20) systems of linear equations, matrices, Gaussian elimination
corresponding sections in Friedberg/Insel/Spence (FIS) --- pp. 26-29 and Sec. 3.4
lecture 2 (01/24/20) vector spaces, subspaces, linear maps
corresponding sections in FIS --- Sections 1.2, 1.3 and 2.1
lecture 3 (01/27/20) subspaces, linear maps, linear independence, spanning, basis
FIS --- Sections 1.5, 1.6, and 2.1
lecture 4 (01/29/20) 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 (01/31/20) A subspace of a finite dimensional vector space is finite dimensional.
lecture 6 (02/03/20) A basis of a subspace can be extended to a basis of the whole space. Sums of subspaces. Linear maps. Range, null space.
lecture 7 (02/05/20) Rank/nullity ("dimension") theorem. Null space measures injectivity. Rank and nullity of a matrix. Some consequences of the rank/nullity theorem.
lecture 8 (02/07/20) Null space measures injectivity. Some consequences of the rank/nullity theorem. Linear isomorphisms.
lecture 9 (02/10/20) Linear bijections have linear inverses. Bases define coordinates. A linear map is uniquely determined by what it does to a basis, isomorphisms send bases to bases.
lecture 10 (02/12/20) Linear maps and matrices, composition of linear maps and matrix multiplication.
lecture 11 (02/14/20) Invertible matrices, change of bases.
lecture 12 (02/17/2020) elementary row operations and elementary matrices
lecture 13 (02/19/2020) (reduced) row echelon form of a matrix, rank and nullity of a matrix in terms of pivots.
lecture 14 (02/21/2020) Space of solutions of a system of linear equations. Quotient vector spaces.
lecture 15 (02/24/2020) Dimension of a quotient vector space. First isomorphism theorem, rank/nullity from first isomorphism, second isomorphism theorem.
Wikipedia page on quotient vector spaces . Notes on quotient vector spaces.
lecture 16 (02/26/2020) Second isomorphism theorem. Dual vector spaces. Dual maps. Transpose of a matrix.
lecture 17 (02/28/2020) Double duals, coordinates in terms of dual bases, multilinear maps, alternating maps.
lecture 18 (03/02/2020) Alternating multilinear maps, the group of permutations.
review problems for the first midterm. Please report any issues/typos.
(03/06/2020) first midterm
lecture 19 (03/09/2020)Any permutation is a (non-unique) product of transpositions.
lecture 20 (03/11/2020) sign of a permutation, inversion, construction of the determinant.
lecture 21 (03/13/2020) properties of determinants: determinant of the transpose, determinants and row operations, determinants of upper triangular matrices...
lecture 22 (03/23/2020) A matrix is invertible if and only if it has nonzero determinant, cofactors,formula for an inverse of a matrix in terms of cofactors.
Video for lecture 22. Read the notes first! The video is fast, and there is a glitch at minute 15 that I don't know how to fix.
lecture 23 (03/25/2020) Similar matrices, determinant of a linear map, trace, definition of eigenvalues and eigenvectors.
Video for lecture 23.
lecture 24 (03/27/2020) Characteristic polynomials. Roots and multiplicities. (Algebraic) multiplicities of eigenvalues.
Video for lecture 24.
lecture 25 (03/30/2020)the product of eigenvalues is determinant, the sum of eigenvalues is trace, a sufficient condition for diagonalizability of a matrix.
Video for lecture 25
review problems (04/01/2020) for the second midterm.tex source
second midterm exam (04/03/2020).
lecture 26 (04/06/2020) Inner products, linear and anti-linear maps, complex conjugate vector spaces, norms, Cauchy-Schwarz inequality, triangle inequality. Video 1 Video 2
lecture 27 (04/08/2020) A collection of orthogonal vectors is linearly independent. Orthogonal and orthonormal bases. Gram-Schmidt.
Video for lecture 27.
lecture 28 (04/10/2020) new version with extra comment to clarify 28.1. Projections and orthogonal projections. Video for lecture 28.
lecture 29 (04/13/2020) adjoint operators and adjoint matrices. Video for lecture 29.
lecture 30 (04/15/2020) Least squares problem, orthogonal projections and adjoint matrices. Isometries. Video for lecture 30.
lecture 31 (04/17/2020) Isometries. Unitary maps. Unitary and orthogonal matrices. Determinants and eigenvalues of unitary matrices. Unitary equivalence. Video for lecture 31.
lecture 32 (04/20/2020) 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. Video for lecture 32.
lecture 33 (04/22/2020) Spectral theorem for matrices. Definition of Jordan normal form. Evaluating polynomials on linear maps. Video for lecture 33.
lecture 34 (04/24/2020) Algebras, homomorphisms(of algebras), ideals, minimal polynomials, Cayley-Hamilton (statement), algebraic and geometric multiplicities of eigenvalues. Video for lecture 34.
lecture 35 (04/27/2020) Direct sum decomposition into generalized eigenspaces. Dimension of a generalized eigenspace is the algebraic multiplicity of the eigenvalue. Video for lecture 35.
lecture 36 (04/29/2020) a proof of existence of Jordan Normal Form and a proof of Cayley-Hamilton for nilpotent linear maps. Video for lecture 36.
lecture 37 (05/01/2020) a proof of existence of Jordan Normal Form and a proof of Cayley-Hamilton in full generality; example of computation of JNF. Video for lecture 37.
lecture 38 (05/04/2020) Another example of computing JNF. A more abstract version of JNF and its application to ODEs.Video for lecture 38.
review problems (05/06/2020) tex file Please report any issues.
Last modified: Oct 14, 14:17:07 CDT 2024