## Fall 2015; Math 417 Course Notes

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.