A lecture on multilinear algebra in June 1998
Rotation of a vector field

1. Tensor Products
 Multilinear mappings of vector spaces
 Existence and universal property of the tensor product
 Commutativity and associativity of the tensor product
 The tensor product in terms of coordinates
 Tensor products and spaces of linear mappings
 Kronecker product of linear mappings (matrices)
 Contraction
 Lowering and raising of indices
 Duality
2. Tensor Algebras
 Covariant, contravariant and mixed tensors
 Classical definition and notation of a tensor in terms of coordinates
 Structure tensor of an algebra
 Mixed tensor algebra
 Universal property of the tensor algebra
3. Exterior Algebras
 Exterior powers and pvectors
 Grassmann coordinates of subspaces
 Alternation operator
 Exterior powers of linear mappings
 Exterior algebra
 Duality and pforms
 Exterior algebra
 Decomposable pvectors
4. Clifford Algebras
 Quadratic forms
 Clifford mappings
 Clifford mappings and exterior algebra
 Clifford Algebras
 Dimension of a Clifford algebra
 Examples of Clifford algebras
 Structures on a Clifford algebra
5. Spinors
 Clifford groups and spin groups
 Isometries
 The centre of a Clifford algebra
 Semisimple modules and representation of algebras
 Spaces of spinors
