| Date | Topic | Sections in the book "Linear Algebra with Applications" by Otto Bretscher, 5th edition | Lecture Notes |
|---|---|---|---|
| 08/24/2021 | Linear systems, solution sets and their geometric interpretation in 2D and 3D; Equivalent linear systems. | 1.1 | Lecture 1 - August 24 |
| 08/26/2021 | Number of solutions for a linear system; Coefficient and augmented matrices; Elementary row operations; (Reduced) Row Echelon Form; Basic and free variables. | 1.2, 1.3 | Lecture 2 |
| 08/31/2021 | Bringing a matrix into (R)REF; Rank of a matrix. | 1.2, 1.3 | Lecture 3 |
| 09/02/2021 | Vectors and operations with them; Vector and matrix form of equation; Homogeneous equation. | 1.2, 1.3 | Lecture 4 |
| 09/07/2021 | Map defined by a matrix multiplication; Interpretation of solving a linear system; Image and Kernel; Onto/surjective and one-to-one/injective maps given by a matrix multiplication. | 2.1, 3.1 | Lecture 5 |
| 09/09/2021 | Linear transformation; Standard matrix. | 2.1, 2.2 | Lecture 6 |
| 09/14/2021 | Examples of linear transformations. Linear combinations. | 2.2, 1.3, 3.1 | Lecture 7 |
| 09/16/2021 | Example of being a linear combination; Spans. | 1.3, 3.1 | Lecture 8 |
| 09/21/2021 | Linear dependence and linear independence. | 3.2 | Lecture 9 |
| 09/23/2021 | Matrix operations; The inverse of a linear transformation. | 1.3, 2.3, 2.4 | Lecture 10 |
| 09/28/2021 | Linear subspace of R^n. | 3.2 | Lecture 11 |
| 09/30/2021 | Basis; Dimension of a subspace; Coordinates. | 3.2, 3.3 | Lecture 12 |
| 10/05/2021 | The basis theorem; Column space of a matrix; Show that null space of a matrix is a subspace. | 3.3 | Lecture 13 |
| 10/07/2021 | Midterm 1 | ||
| 10/14/2021 | Basis of a null space; Rank-Nullity theorem; Some operations with matrices: powers and transpose; Orthogonality, length, unit vectors. | 3.3, 5.1 | Lecture 14 |
| 10/19/2021 | Orthogonal and orthonormal bases; Orthogonal complement. | 5.1 | Lecture 15 |
| 10/21/2021 | Projection on a line; Orthogonal projection on a subspace; Orthogonal matrix; Introduction to Gram-Schmidt method. | 5.1, 5.2, 5.3 | Lecture 16 |
| 10/26/2021 | Gram-Schmidt method; Change of basis. | 5.2, 3.4 | Lecture 17 |
| 10/28/2021 | Change of basis; A matrix of a linear transformation. | 3.4 | Lecture 18 |
| 11/02/2021 | Determinant of a matrix; the matrix of minors and the matrix of cofactors. | 6.1, 6.2 | Lecture 19 |
| 11/04/2021 | Find the inverse of a matrix using the matrix of cofactors; Geometric interpretation of determinant; Computing of the determinant using row operations. | 6.2, 6.3 | Lecture 20 |
| 11/09/2021 | Eigenvalues and eigenvectors and how to find them. | 7.2, 7.3 | Lecture 21 |
| 11/11/2021 | Diagonalization | 7.1 | Lecture 22 |
| 11/16/2021 | Continue on diagonalization; Definition of a symmetric matrix | 7.1, 8.1 | Lecture 23 |
| 11/18/2021 | Midterm 2 | ||
| 11/23/2021 | Orthogonal diagonalization of symmetric matrices; | 8.1 | Lecture 24 |
| 11/30/2021 | Introduction into Singular Value Decomposition | 8.3 | Lecture 25 |
| 12/02/2021 | Singular Value Decomposition | 8.3 | Lecture 26 |
