The tensor product, $\mathcal{V}\otimes\mathcal{W},$ of a pair of [vector spaces](Vector%20spaces.md) $\mathcal{V}$ and $\mathcal{W}$ is a vector space that contains [linear combinations](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#linear%20combinations) of products of vectors $v_i \in \mathcal{V}$ and $w_j\in\mathcal{W}$ expressed as $a_{11}(v_1\otimes w_1)+a_{12}(v_1\otimes w_2)+a_{21}(v_2\otimes w_1)+a_{22}(v_2\otimes w_2)...$ The vectors $v_i\otimes w_j$ are referred to as tensor products. %%This way of defining it is from Altland and von Delft pg. 148.%% # Properties of tensor products of vector spaces #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra