MAU11100 Linear algebra

11+11 weeks of teaching with 3 lectures and 1 tutorial per week.

On successful completion of this module, students will be able to

Apply various standard methods to solve systems of linear equations.
Compute the sign of a given permutation and apply known results to compute the determinant of a given square matrix.
Demonstrate that a given set of vectors forms a basis for a vector space, compute the coordinates of a given vector relative to this basis, and find the matrix of a linear operator with respect to this basis.
Give examples of sets where some of the defining properties of vectors, matrices, vector spaces, subspaces and linear operators fail.
Find an explicit basis for the null space of a given matrix.
Solve linear recursive relations involving two or more terms.
Apply standard techniques to obtain the Jordan form and a Jordan basis for a given complex square matrix.
Compute the matrix of a bilinear form with respect to a given basis.
Apply various standard method to determine the signature of a given symmetric bilinear form.
Combine various results established in the module to either prove or disprove statements involving concepts introduced in the module.

Lines, planes and vectors, dot and cross product.
Linear systems, Gauss-Jordan elimination, reduced row echelon form.
Matrix multiplication, elementary row operations, inverse matrix.
Odd and even permutations, determinants, transpose matrix.
Minors, cofactors, adjoint matrix, inverse matrix, Cramer's rule.
Vector spaces, linear independence and span, basis and dimension.
Linear operators, matrix of a linear operator with respect to a basis.
Diagonalisation: recursive relations, diagonalisable matrix, eigenvalues, eigenvectors, characteristic polynomial, null space, nullity.
Jordan forms: generalised eigenvectors, column space, rank, direct sum, invariant subspace, Jordan chain, Jordan block, Jordan form, Jordan basis, similar matrices, minimal polynomial.
Bilinear forms: matrix of a bilinear form, positive definite, symmetric, inner product, orthogonal and orthonormal basis, orthogonal matrix, quadratic form, signature, Sylvester's criterion.