| Week 1 |
Introduction. Definition of a vector space. Examples (column vectors, matrices, functions). First properties (cancellation law, various uniquenesses,...) |
| Week 2 |
Definition of a subspace. Examples. Linear combinations. Definition of a basis. Spanning sets and linearly independent sets. |
| Week 3 |
Extraction of a basis from a spanning set. Definition of the dimension. The replacement lemma. Bases and dimension. Cardinality of spannning and linearly independent sets. Subspaces of finite dimensional spaces. |
| Week 4 |
Defintion of a linear transformation. Elementary properties. Examples. The set of linear maps as a vector space. Range and kernel. The dimension theorem. Example. |
| Week 5 |
p x n matrices. The matrix product. Properties. The matrix product is not commutative. Examples. Traces. Invertible matrices. The matrix of a linear transformation w.r.t. two bases. |
| Week 6 |
Linear transformations and matrices: Examples. The composition of linear maps and the matrix product. Change of bases. |
| Week 7 |
Systems of linear equations. Coefficients and augmented matrices. Row reduction. The echelon and reduced echelon forms. Consequences: Existence and uniqueness of solutions. The rank of a matrix. |
| Week 8 |
More on row reduction. Examples. Invertibility & existence and uniqueness of solutions. The general solution of inhomogeneous systems. The determinant: introduction and first properties. |
| Week 9 |
The determinant and invertibility. Computing the determinant by column reduction. The determinant of a product. Existence: The cofactor expansion. Examples. Eigenvalues and eigenvectors: definition and first examples. |
| Week 10 |
Diagonalizable linear maps and matrices. Characterization of the eigenvalues. The characteristic polynomial and its roots. Computing eigenvalues. Eigenspaces. Algebraic and geometric multiplicities, and relation to diagonalizability. |
| Week 11 |
More on diagonalization, examples. Inner product spaces: Introduction and definitions. The norm. Examples. The Cauchy-Schwarz inequality. |
| Week 12 |
Proof of Cauchy-Schwarz. The triangle inequality and Pythagoras' theorem. Orthonormal bases. The adjoint of a linear map. Self-adjoint linear transformations and matrices. The spectral theorem for self-adjont matrices. Google Page Rank. |
| Week 13 |
Recap: inner products, norms, adjoints of maps and matrices. Orthogonal and unitary transformations. The spectral theorem rephrased. Examples. General overview of the course. |