Tensor product

The tensor product of modules or vector spaces is an object that represents bilinear maps as linear maps. It combines inputs while preserving the relations imposed by their scalars.

Bilinear map: A function of two arguments that is linear in each argument when the other is fixed. Tensor products are defined so bilinear maps out of two modules correspond to linear maps from their product.
Pure tensor: An element of a tensor product written as the tensor of one element from each factor. Pure tensors generate the product, though sums of them may be needed for general elements.
Matrix multiplication: The operation that composes linear transformations by summing products of matrix entries. Matrix multiplication can be represented using tensor contraction and products of component spaces.
Direct sum: A construction that combines objects as independent components with coordinatewise addition and scalar action. Unlike a tensor product, a direct sum combines inputs linearly rather than encoding bilinear interactions.
Flat module: A module whose tensor product functor preserves exact sequences. Flatness measures whether tensoring with a module preserves algebraic relations without creating new kernels.
Universal property: A property that characterizes an object through a unique map satisfying specified conditions. It gives the defining correspondence between bilinear maps and linear maps from a tensor product.
Tensor contraction: An operation that pairs tensor indices using a bilinear form or duality and removes the paired indices. Contraction turns tensor products into maps or scalars by pairing selected factors.
Kronecker product: A block matrix formed from two matrices by multiplying every entry of one by the entire other matrix. In chosen bases, the tensor product of linear maps is represented by the Kronecker product.
Cartesian product: The set of ordered tuples drawn from specified sets, with no additional identifications by scalar relations. The underlying pairs alone do not impose the bilinear relations that define a tensor product.
Tor functor: A family of functors measuring the failure of tensor product to preserve exactness. Tor detects the homological information lost when tensoring is not exact.

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