September 9: Class Policy. Gauss elimination
September 11: Sets of solutions of systems of linear equations. Elementary row operations. Reduced row echelon form and its uniqueness.
September 14: Reduced row echelon form and its uniqueness. Vector spaces. Linear combinations. Homogeneous systems of linear equations.
September 16: Quiz. Examples of vector spaces (=linear spaces). Vector subspaces.
September 18: Matrix multiplication. Identity and inverse matrix. Finding matrix inverse using reduced row echelon form.
September 21: Proving that the (A|I) -> RREF algorithm finds a two-sided inverse of a square matrix, or there is no inverse. Uniqueness of the inverse.
September 23: Quiz. The rank of a matrix. The left or right inverse of a matrix..
September 25: Mappings. Surjective, injective, bijective mappings, inverse, left and right inverse.
September 28: The matrix of a linear transformation. Examples..
October 2: Onto and injective linear transformations. Back to vector spaces: linear independent and spanning sets..
October 5: Lineadly independent and spanning sets II. Bases. Base change matrix.
October 7: Quiz. Base change matrix II..
October 9: Basis of a column space and a solution space. Matrix of a linear map in any two bases.
October 12: The matrix of a linear transformation and base change. The determinant of a matrix. Even and odd permutations..
October 14: Quiz. Even/odd permutations and reversed pairs. Determinants and row operations.
October 16: Linearity of the determinant in one row or column.Calculating determinants by Gauss elimination. det(AB)=deta(A)det(B).Cramer's rule for linear equations and inverse matrix.
October 21: Quiz. Row and column expansion formulas for determinants. Similar matrices. Determinant of a linear transformation with the same domain and codomain.
October 23: Eigenvalues, eigenvectors and diagonalization..
October 26: Complex eigenvales. Fields. Linear algebra in complex numbers and circuits with resistors, capacitors, inductors.
October 30: Degenerate eigenvalues. Algebraic and geometric multiplicity. Jordan blocks.
November 2: Jordan canonical form.
November 4: Symmetric matrices. Orthonormal bases and their orientation.
November 6: Orthogonal matrices. Principal axes. The case of degenerate eigenvalues..
November 9: Orthogonal row echelon form. Gramm-Schmidt orthogonalization process..
November 11: Centering a quadric. Hermitian matrices..
November 13:Principal axes for Hermitian matrices. Examples..
November 16:The principal axes theorem for Hermitian matrices. Orthogonal projection formula..
November 18:Gramm determinant and volume. POsitive definite matrices. Projection onto the orthogonal complement..
November 20:Application: Pearson chi squared test. Singular values.
November 23:Interpretation of singular values. Application of Hermitian matrices: Quantum mechanics.
November 25:Questions and answers.
November 30: Multivariable substitution in integrals. Exterior algebra..
December 2: Differential forms
December 4: Exterior differential. Stokes' theorem. HOdge star-operator.
December 7: Maxwell-Lorentz equations using differential forms in space-time. Special relativity.
December 9: Topics for Test 3. Review Part 1.
December 11:Review Part 2.