The tensor algebra of a vector space is the graded algebrawith multiplication given by concatenation of tensors.
The tensor square is the second tensor power . Over a field of characteristic different from two it decomposes as the direct sum of the symmetric square and exterior square.
The symmetric algebra is the quotient of by the ideal generated by . It is graded by symmetric powers:
The nth symmetric power is the quotient of that identifies tensors differing by a permutation of their factors. Equivalently, over characteristic zero it is the subspace of symmetric tensors.
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Tensor algebra is a mathematical framework that extends the concepts of linear algebra to accommodate tensors, which are multi-dimensional arrays that generalize scalars, vectors, and matrices. In simpler terms, tensors can represent data in more complex ways compared to traditional linear algebra structures. ### Key Concepts in Tensor Algebra: 1. **Tensors**: - A scalar is a 0th-order tensor. - A vector is a 1st-order tensor.