Tuesday, Aug 30, 05: Solution(s) of upper-triangular systems by backward substitution, examples on inconsistent systems and on systems with infinitely many solutions, equivalent systems, definition of determinant, how to find the determinant of a square diagonal, lower-triangular, or upper-triangular matrix, the relationship between the determinant and singularity of matrices, elementary row operations, elementary matrices.
Wednesday, Aug 31, 05: Row echelon form, Gaussian elimination, how to use Gaussian elimination to find the determinant of a square matrix.
Friday, Sep 2, 05: Continuation.
Tuesday, Sep 6, 05: Gauss-Jordan elimination, how to use Gauss-Jordan elimination to find the determinant of a square matrix, determinants of 2 by 2 matrices, inverse of nonsingular 2 by 2 matrices.
Wednesday, Sep 7, 05: How to use Gauss-Jordan elimination to find the inverse of a nonsingular matrix.
Friday, Sep 9, 05: pivoting, homogeneous systems, theorems related to linear systems, determinants and inverses.
Monday, Sep 12, 05: row-equivalent systems, theorems, determinants (Chapter 3).
Tuesday, Sep 13, 05: continue determinants, theorems.
Wednesday, Sep 14, 05: quiz, continue theorems, the adjoint.
Friday, Sep 16, 05: using the adjoint to find the inverse, using the determinant to find the solution of a linear system (Gramer's rule), orthogonal matrices, and other things.
Monday, Sep 19, 05: concluding remarks of Chapter 3, introduction to Chapter 6.
Tuesday, Sep 20, 05: Section 6.1 (real vector spaces).
Wednesday, Sep 21, 05: continue with 6.1.
Friday, Sep 23, 05: Finish 6.1, begin 6.2 (subspaces).
Monday, Sep 26, 05: Exam I.
Tuesday, Sep 27, 05: subspaces continued.
Wednesady, Sep 28, 05: subspaces continued, linear combination, span.
Friday, Sep 30, 05: begin Section 6.3, span.
Monday, Oct 3, 05: continue with spann
Tuesday, Oct 4, 05: Continue span, theorems, span for nullspace.
Wednesday, Oct 5, 05: Linear dependence and linear independence.
Friday, Oct 7, 05: continue with linear dependence/independence; theorems, begin 6.4 bases.
Monday, Oct 10, 05: Bases continued, theorems and definitions, dimension, how to determine if a set is a basis, how to write a vector as a linear combination of the basis
Tuesday, Oct 11, 05: continue with examples, how to find a basis from set S for span S.
Monday, Oct 17, 05: how to find a basis for the subspace of all vectors of a given form, how to expand a linearly independent set to a basis.
Tuesday, Oct 18, 05: more examples, basis for nullspace (6.5), nullity.
Wednesday, Oct 19, 05: more examples on 6.4 and 6.5.
Friday, Oct 21, 05: Section 6.6 (row space, row rank, column space, column rank, rank).
Monday, Oct 24, 05: 6.6 continued.
Tuesday, Oct 25, 05: Section 6.7 (Coordinates and change of basis), coordinates, transition matrix, ordered basis, ordered basis notation, how to change from an order basis to the stndard basis and vice versa.
Wednesday, Oct 26, 05: How to change from any ordered basis to any other ordered basis and the corresponding transition matrix, applications to R^n.
Friday, Oct 28, 05: Quiz, change of basis in P_n, vector norms.
Monday, Oct 31, 05: Orthogonal vectors and basis, orthonormal vectors and basis, Gram-Schmidt orthogonalization process.
Friday, Nov 4, 05: Finish Section 6.8.
Monday, Nov 7, 05: begin Chapter 10 (Linear Transformations): definition of linear transformation, domain, target, range, image, preimage, image of a set, preimage of a set, one-to-one, onto, kernel, nullity, rank.
Tuesday, Nov 8, 05: Related theorems, examples on how to prove a function is a linear transformation or not, on how to use the values of a linear transformation L at a basis to find L at any other element of the domain, on how to find the kernel and a basis for it and a basis for the range, and on how to find the nullity and the rank.
Wednesday, Nov 9, 05: Continue with similar examples from R^n and P_n and examples on determine whether an element from the domain belongs to the kernel and on whether an element from the target belongs to the range, etc.
Friday, Nov 11, 05: More examples.
Monday, Nov 14, 05: Exam II (Material: Chapter 6 and the related material we covered).
Tuesday, Nov 15, 05: The Matrix of a linear transformation.
Wednesday, Nov 16, 05: reflections about x-axis, y-axis, origin, y=x, y=-x, rotations clockwise and counterclockwise, contractions, stretching, projections, etc. How to find the matrices of such linear transformations, and how to find their formulas. Thus, Chapter 10 is finished.
Friday, Nov 18, 05: Complex numbers, solving equations which have complex solutions, roots of unity, unitary, matrix conjugate, Hermitian transpose, Hermitian, and skew-Hermitian matrices.
Monday, Nov 28, 05: Begin Chapter 8 (eigenvalues, eigenvectors, and diagonalization): define eigenvalues, eigenvectors, eigenpairs, spectrum, characteristic polynomial, characteristic equation, facts (sum of eigenvaleus=trace and product of eigenvalues=determinant, matrix is singular iff 0 is an eigenvalue), characteristic equation of 2x2 matrices.
Tuesday, Nov 29, 05: examples.
Wednesday, Nov 30, 05: examples continued, geometric multiplicity, algebraic multiplicity, defective matrices, non-defective matrices, facts (eigenvalues of diagonal/lower-triangular/upper-triangular are diagonal elements, geom multip <= algebraic multip, eigenvalues A equal eigenvalues its transpose, eigenvalues AB = eigenvalues BA, eigenvalues of inverse of invertible A = recipricals eigenvalues A and eigenvectors of them are same), examples on defective matrices, on eigevalues with geometric multiplicity < algebraic multiplicity, and on eigenvalues with geometric multiplicity = algebraic multiplicity.
Friday, Dec 2, 05: more examples, if B is row-equivalent to A, do they have same eigenvalues? Define similar matrices diagonalizable matrices, diagonalizable and defective relationship, facts about diogonalizable matrices, how to diagonalize a diagonalizable matrix, etc.
Monday, Dec 5, 05: state and prove relationship between the eigenvalues and eigenvectors of two similar matrices, examples.
Tuesday, Dec 6, 05: Simple eigenvalues, theorems about diagonalization, theorems about similar matrices, eigenvalues of Hermitian, skew-Hermitian, and unitary matrices, example on a real matrix with complex eigenvectors and real eigenvalues.
Wednesday, Dec 7, 05: eigenvalues of powers of matrix A, eigenvalues occuring in conjugate paires, spectral radius, matrix norms (one, two, and infinity).
Friday, Dec 9, 05: Companion matrix, right and left eigenvectors, block matrices.