Familiar vector spaces -- Gaussian elimination -- Two hundred years of algebra -- Computational matters -- Detached coefficients -- Another fifty years -- A little geometry -- Geometric vectors -- Higher dimensions -- Euclidean distance -- Geometry, plane and solid -- The algebra of square matrices -- The summation convention -- Multiplying matrices -- More algebra for square matrices -- Decomposition into elementary matrices -- Calculating the inverse -- The secret life of determinants -- The area of a parallelogram -- Rescaling -- 3 x 3 determinants -- Determinants of n x n matrices -- Calculating determinants -- Abstract vector spaces -- The space Cn -- Abstract vector spaces -- Linear maps -- Dimension -- Image and kernel -- Secret sharing -- Linear maps from Fn to itself -- Linear maps, bases and matrices -- Eigenvectors and eigenvalues -- Diagonalisation and eigenvectors -- Linear maps from C2to itself -- Diagonalising square matrices -- Iteration's artful aid -- LU factorisation -- Distance preserving linear maps -- Orthonormal bases -- Orthogonal maps and matrices -- Rotations and reflections in R2and R3 -- Reflections in Rn -- QR factorisation -- Diagonalisation for orthonormal bases -- Symmetric maps -- Eigenvectors for symmetric linear maps -- Stationary points -- Complex inner product -- Cartesian tensors -- Physical vectors -- General Cartesian tensors -- More examples -- The vector product -- More on tensors -- Some tensorial theorems -- A (very) little mechanics -- Left-hand, right-hand -- General tensors -- General vector spaces -- Spaces of linear maps -- A look at L(U, V) -- A look at L(U, U) -- Duals (almost) without using bases -- Duals using bases -- Polynomials in L(U, U) -- Direct sums -- The Cayley-Hamilton theorem -- Minimal polynomials -- The Jordan normal form -- Applications -- Vector spaces without distances -- A little philosophy -- Vector spaces over fields -- Error correcting codes -- Vector spaces with distances -- Orthogonal polynomials -- Inner products and dual spaces -- Complex inner product spaces -- More distances -- Distance on L(U, U) -- Inner products and triangularisation -- The spectral radius -- Normal maps -- Quadratic forms and their relatives -- Bilinear forms -- Rank and signature -- Positive definiteness -- Antisymmetric bilinear forms -- Further exercises.