Linear Algebra

1 Linear Algebra

Linear algebra studies vector spaces and the linear maps between them. The material is organised into the following notes.

1. Vector Spaces - fields and vectors, linear combinations, span, independence, subspaces, basis and dimension.
2. Linear Maps - linear maps, their matrices, change of basis, image, kernel and rank-nullity.
3. Matrices - arithmetic, transpose, special matrices, row reduction, determinant, inverse, trace, rank and positive definiteness.
4. Eigenvectors and Eigenvalues - eigenvalues, characteristic polynomial and diagonalisation.
5. Inner Product Space - bilinear forms, inner products, Cauchy-Schwarz and the induced norm.
6. Orthogonality and Projections - orthogonality, Gram-Schmidt and projections.
7. Matrix Decompositions - LU, QR, Cholesky, spectral decomposition and SVD.

Norms and metrics on vector spaces are covered in Normed Space and Metric Spaces. Gradients, level sets and the Jacobian are covered in Vector Calculus.

Backlinks

• Affine Function
• Algebra
• Conditional Expectation
• Jacobian and Hessian Matrices
• Mathematics
• Probability Theory