## Description

- Vectors: inner products, norms, main operations (average, standard deviation, ...)
- Matrices: matrix-vector and matrix-matrix multiplication, Frobenius norm, Complexity, sparsity
- Special matrices: Diagonal, Upper Triangular, Lower triangular, Permutation (general pair), inverse and orthogonal
- A square and invertible: LU decomposition (aka gaussian elimination), LU-P decomposition, Cholesky decomposition
- Ax=b via LU-P decomposition: forward and backward substitution
- (sub)Vector spaces: definitions, span, bases (standard, orthogonal, orthonormal), dimension, direct sum, orthogonal complement, null space, orthogonal complement theorem
- Gram-Smith orthogonalization and QR decomposition (square and invertible A, general non-square)
- Ax=b via QR decomposition. LU-P vs QR

- Linear maps: image space, kernel, column and row rank
- Fundamental Theorem of Linear Algebra (Part I): rank-nullity Theorem, the 4 fundamental subspace
- Eigenvalues/eigenvector and Shur decomposition
- Projection matrices: oblique and orthogonal, properties
- Positive semidefinite matrices: properties and quadratic functions square root matrix
- Properties of A'A and AA' and Polar decomposition
- Singular Value Decomposition: proofs and properties
- Pseudo-inverse: definition and relation to SVD
- Fundamental Theorem of Linear Algebra (Part II): special orthogonal basis for diagonalization
- Least-Squares: definition, solution and algorithms
- Ill-conditioned problems vs stability of algorithms, numerical conditioning of algorithms, numerical conditioning