A Brief Introduction to Tensors∗

In general, a tensor is a multilinear transformation defined over an underlying finite dimensional vector space. In this brief introduction, tensor spaces of all integral orders will defined inductively. Initially the underlying vector space, V, will be assumed to be an inner product space in order to simplify the discussion. Subsequently, the presentation will be generalized to vector spaces without inner product. Usually, bold-face letters, a, will denote vectors in V and upper case letters, A, will denote tensors. The inner product on V will be denoted by a · b.