Loughborough University (LU)

Notes defining Euclidean Vector Spaces, Gram Matrices and Orthogonal Matrices; with examples of each. Also includes a proof of the Cauchy-Bunyakovsky-Schwarz Inequality.

Notes defining Bilinear Forms, Matrices of Bilinear Forms and Coordinate Representations. Covers concepts such as Rank, Symmetric and Skew-Symmetric Matrices, and Congruence.

Notes on solving Cauchy problems using matrix concepts such as matrix powers, diagonalisability and eigenspaces. Includes various examples including the Fibonacci sequence.

Notes discussing theorems for quadratic forms on ℝ vector spaces including rigorous proofs.

Notes defining and giving examples of linear maps. Also includes elementary operations for linear maps as well as introducing the concept of Kernels and Images and the Rank-Nullity Formula for linear maps.

Notes on how to find the eigenvalues and eigenvectors of a matrix as well as how to determine whether a matrix is diagonalisable.

Notes covering the rules of matrix algebra including the elementary operations, transposition and inverting a matrix.

Notes on how to find the determinant of an nxn matrix. Includes a number of theorems related to the determinant of a matrix as well as the effects of the determinant under elementary operations.

Definitions of rank and minors of a matrix with all associated theorems needed for this section of Linear Algebra 1.

Notes on the Cayley-Hamilton Theorem in Linear Algebra with its corollary and a rigorous proof.