| Week | Date | Contents |
|---|---|---|
| 1 | 0514 |
First lecture outline: course homepage, textbook, grading, homework and exam policies The rest of the lecture was cancelled due to a power disruption |
| 0516 |
§1.1 Systems of linear equations
definitions, Ex 1: a system of 2 equations of 2 unknowns; geometric interpretation, possibilities of the solution set. Matrix notation, Ex 2: redo Ex 1 in matrix form. |
|
| 0517 | elementary row operations, Ex 3, forward and backward phases, Ex
4 and Ex 5 §1.2 Row reduction and echelon forms definitions of echelon form and reduced echelon form, pivot position and pivot column, Ex 1, 2; Finding general solution from reduced echelon form, Ex 2 again, Deciding existence and uniqueness from echelon form. |
|
| 0518 | Ex 3 §1.3 Vector equations vectors, sum and scalar multiple, Ex 1, geometric meaning in R^2, linear combinations, Ex 2, 3 and 4, equivalence of being a linear combination and the linear system, Span, Ex 5, 6. §1.4 Matrix equations product of a matrix and a vector, Ex 1 and 2. Equivalence of a matrix equation, a vector equation and a linear system, Ex 3. When does a matrix equation Ax=b has a solution for a given b? Ex 4. When does a matrix equation Ax=b has a solution for every b? |
|
| 2 | 0521 |
Victoria Day |
| 0523 |
H1 due Ex 5. Properties and computation of Ax, Ex 6 §1.5 Solution sets of linear systems solution sets of homogeneous systems, Ex 1, 2, 3 |
|
| 0524 | Ex 3 solution 2, the solution set of a homogeneous system is a
span. Nonhomogeneous system, Ex 4 and 5, properties of the solution set of a
nonhomogeneous system, geometric meaning. (We skip §1.6) §1.7 Linear independence definition, Ex 1, 2, special cases, Ex 1 again, Ex 3, equivalence of linear dependence and existence of redundant vector, Ex 2 again, Ex 4, p>m implies linear dependence, Ex 5 §1.8 Linear transformations terminologies, matrix transformations, Ex 1 |
|
| 0525 | Ex 2-4, Ex 5 on gemetric meaning of 2x2 matrix transformations.
Linear transformations, definition and properties, Ex 6-9. §1.9 the matrix of a linear transformation standard basis, the theorem that any linear transformation from R^n to R^m is a matrix transformation, Ex 1-3 |
|
| 3 | 0528 |
Ex 4-5, definition and theorem of onto and one-to-one linear
transformations, Ex 6. §2.1 Matrix operations sum and scalar multiplication of matrices, Ex 1, composition of linear transformations, definition and formula of matrix multiplication, Ex 2, 3, computation rule, Ex 3 again, properties of matrix multiplication, Ex 4. Powers of a square matrix, Ex 5, 6. |
| 0530 |
Midterm exam 1 | |
| 0531 | transpose, Ex 7, 8, 9 §2.2 The inverse of a square matrix definition of invertible/inverse. Inverse is unique. Formula of the inverse matrix when n=2. Ex 1. Application for solving linear system, Ex 2. Properties of inverse matrix, Ex 3. Algorithm using row reduction, Ex 4 and 5. Right and left inverse for non-square matrices, Ex 6**. |
|
| 0601 | §2.3 Characterization of inverse
matrices Invertible Matrix Theorem: on equivalent statements for invertible square matrices, Ex 1 and 2**, geometric meaning. Note: The exams will not include elementary matrices, right and left inverses. We also skip 2.4 §2.5 Subspaces of R^n definition of subspaces, Ex 1, 2, 3. Column and null spaces of a matrix, Ex 4. Spanning set and basis. Ex 5--8. |
|
| 4 | 0604 |
The theorem that pivot columns form a basis of the column
space. Ex 9. §2.6 Dimension and rank Coordinate vector, Ex 1. Dimension theorem and definition, Ex 2. Ex 3. Rank. The rank theorem. Ex 4. The basis theorem. Extension of the Invertible Matrix Theorem. §3.1 Introduction to determinants Determiniant of a 3x3 matrix, Ex 1. Cofactor. Determiniant of an nxn matrix. The theorem on cofactor expansion. Ex 1 again, Ex 2. determinant of a triangular matrix, Ex 3. Ex 4. |
| 0606 |
H2 due §3.2 Properties of determinants theorem on the change of the determinant under row/column operations, proof for 2x2 matrices, Ex 1--4. A square matrix A is invertible iff det A is nonzero. Ex 5. det A^T = det A. det(AB) = det A . det B. |
|
| 0607 | Ex 6, determinant as a ratio of volumes (not in
exam). §4.1 Eigenvectors and eigenvalues Ex 1, definition, Ex 2-4. Eigenvalues of triangular matrices, Ex 5. Eigenvectors with distinct eigenvalues are linearly independent §4.2 The characteristic equation The characteristic equation, Ex 1, Ex 2 |
|
| 0608 | Ex 3, algebraic and geometric
multiplicity, Ex 4-5, characteristic eq for triangular matrices.
Discrete dynamical systems, Ex 6 (population dynamics), Ex 7
(Fibonacci's rabbits)
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|
| 5 | 0611 |
Similarity, similar matrices have the same determinants, characteristic
polynomials, eigenvalues, algebraic and geometric multiplicities of eigenvalues, Ex
8-9. A matrix with no real eigenvalue, Ex 10. §4.3 Diagonalization Ex 1, 2. Diagonalizable matrix. An nxn matrix is diagonalizable if and only if it has n linearly independent eigenvectors. Ex 3-7. Special case when all eigenvalues are distinct. Ex 8. Ex 9. General case: An nxn matrix is diagonalizable if and only if the sum of geometric multiplicities is n. |
| 0613 |
Midterm exam 2 | |
| 0614 |
§4.4 Eigenvectors and linear
transformations Matrix of a linear transformation in R^n relative to a basis, Ex 1-4. (We do not consider general linear transformations between vector spaces.) §4.5 Complex eigenvalues Complex numbers, Ex 1-2, Theorem of De Moivre, Ex 3. Link to Appendix B on complex numbers. |
|
| 0615 | Complex eigenvalues, Ex 4-7, conjugate eigenvalue, A=PCP^{-1} with C = [[a,b]^T, [-b, a]^T] §4.6 skipped (It contains more advanced topics in discrete dynamical systems, including graphics and those with complex eigenvalues. However, the limit behavior of the simple discrete dynamical systems studied in §4.2 will be in final exam.) §5.1 Inner product, length, and orthogonality dot product, Ex 1, length, Ex 2-3, distance, Ex 4, orthogonality, Ex 5, orthogonal to a set, orthogonal complement, Ex 6. Theorem on orthogonal complement. |
|
| 6 | 0618 |
H3 due Ex 7, Ex 8, angles, Ex 9 (We skip Theorem 3) §5.2 Orthogonal sets orthogonal sets, Ex 1-2, an orthogonal set of nonzero vectors is linearly indepdent. Formula for coefficient. Ex 3. orthogonal projection onto a line, Ex 4-5 |
| 0620 |
Ex 6 matrix of a reflection. Orthonormal sets, Ex 7. Matrices with orthonormal columns, Ex 8. An mxn matrix U has orthonormal columns iff U^T U=I_n. In that case, U: R^n -> R^m preserves length and angle. Ex 9. | |
| 0621 | last lecture Orthogonal/orthonormal matrices, Ex 10 and remarks §5.3 Orthogonal projections Theorem 8 on orthogonal decomposition, Ex 1. Theorem 9 on best approximation, Ex 2. Projection to a plane with non-orthogonal basis, Ex 3, Ex 4. Review of April 2017 final exam. |
|
| 0622 | no class |