| 1 |
Systems of Linear Equations |
Section 0.1: Systems of linear equations, row equivalence |
| 2 |
Echelon Form |
Section 0.2: Gaussian and Gauss-Jordan elimination, (reduced) row-echelon form, back-substitution |
| 3 |
Matrices |
Section 0.3: Matrices, matrix operations, block multiplication |
| 4 |
Matrices (cont.) |
Section 0.4: Matrices and linear systems, elementary matrices, (reduced) row-echelon matrices |
| 5 |
Solution Spaces |
Section 0.5: The space of solutions to a homogeneous linear system, uniqueness of the reduced row-echelon form, matrix rank, criterion for existence of solutions |
| 6 |
Inverses and Transposes |
Section 0.6: Matrix inverses (right, left), invertible matrices, transpose of a matrix, symmetric matrices |
| 7 |
Fields and Spans |
Section 1.1: Definition of a field F, examples: Q, R, C, Z/pZ (see also 1.6. pp. 132-133), linear combinations of vectors, and spans in Fn |
| 8 |
Vector Spaces |
Section 1.2: Vector spaces, definition and examples, sub-spaces, the row space, column space, and nullspace of a matrix |
| 9 |
Linear Independence |
Section 1.3: Linear independent vectors |
| 10 |
Basis and Dimension |
Section 1.4: Basis of a vector space, dimension, bases for the row space and column space of a matrix, Rank plus nullity theorem for matrices, Basis extension theorem |
| 11 |
Coordinates |
Section 1.5: Coordinates with respect to an ordered basis, change of coordinates matrix |
| 12 |
Review for Quiz 1 |
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| 13 |
Quiz 1 (Chapters 0-1) |
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| 14 |
Determinants |
Section 2.1 (pp. 137-143): Determinant function (definition, properties, uniqueness), computing determinants using row-reduction
Section 2.1 (pp. 144-146): invertible matrices, det(AB) = det(A)det(B), det(At) = det (A) |
| 15 |
Permutations |
Section 2.2: Permutations and the permutation definition of the determinant |
| 16 |
Determinants (cont.) |
Section 2.2: Permutations and the permutation definition of the determinant |
| 17 |
Laplace Expansion |
Section 2.3: Cofactor (Laplace) expansion of the determinant, the adjoint of a matrix, finding the inverse using the adjoint, Cramer's rule |
| 18 |
Review for Quiz 2 |
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| 19 |
Quiz 2 (Chapter 2) |
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| 20 |
Linear Transformations |
Section 3.1: Linear transformations (definition, examples), matrix associated to a linear transformation |
| 21 |
Rank, Kernel, Image |
Section 3.2: Properties of linear transformations, rank, kernel, image, Rank plus nullity theorem for linear transformations, one-one, onto, isomorphism |
| 22 |
Matrix Representations |
Section 3.3: Matrix representations for linear transformations, similar matrices |
| 23 |
Eigenspaces |
Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |
| 24 |
Eigenspaces (cont.) |
Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |
| 25 |
Diagonalization |
Section 3.5: Diagonalizable linear operators and matrices |
| 26 |
Cayley-Hamilton Theorem |
Section 6.1: Cayley-Hamilton theorem, minimal polynomial |
| 27 |
Jordan Canonical Form |
Section 6.4 (pp. 373-376): Jordan form, generalized eigenvectors and Primary decomposition theorem from section 6.5 (see also J. Starr's notes from Fall 2004) |
| 28 |
Review for Quiz 3 |
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| 29 |
Quiz 3 (Chapter 3) |
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| 30 |
Computing Generalized Eigenvectors |
Section 6.4 (pp. 376-384): More on Jordan form, computing generalized eigenvectors |
| 31 |
Norms and Inner Products |
Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |
| 32 |
Norms and Inner Products (cont.) |
Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |
| 33 |
Orthogonal Bases |
Section 4.2: Orthogonal and orthonormal bases, Gram-Schmidt algorithm, QR decomposition |
| 34 |
Orthogonal Projections |
Section 4.3: Orthogonal projections, orthogonal complement, direct sums |
| 35 |
Isometries, Spectral Theory |
Section 4.5 (pp. 282-285): Isometries, orthogonal and unitary matrices |
| 36 |
Singular Value Decomposition |
Section 4.5 (pp. 286-291): Self-adjoint operators, symmetric and hermitian matrices, eigenvalues of self-adjoint operators, Principal axis theorem, Spectral resolution |
| 37 |
Polar Decomposition |
Section 4.6: Singular value decomposition, positive (semi)definite matrices, Polar decomposition |
| 38 |
Review for the Final |
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