Finite Dimensional Vector Spaces

The notion of basis is used to define finite dimensionality. The theorem that every finite-dimensional vector space is linearly isomorphic to some Euclidean space is proved and used to establish that finite-dimensional vector spaces can only have one dimension. Representation of linear transformations by matrices resulting from choice of bases receives careful treatment, and students may practice this representation in the associated software. The important Basis Extension and Spanning Set Reduction theorems are proved. These concepts are used to give a rigorous proof of multilinearity of the determinant. Multilinearity, in turn, is used to prove that cofactor expansion may be used to compute determinants. The determinant is shown to be the unique (normalized) multilinear function defined on square matrices. The row, column, and null spaces of a matrix are defined, and their important properties are proved, including orthogonality. Orthonormal bases and orthogonal matrices are introduced and used to describe the Gram-Schmidt process. The important theorem that a subspace and its orthogonal complement span the entire space is proved and used to define orthogonal projection of a vector onto a subspace.