• Mathematics

    The Tensor Product: from vector spaces to categories

    Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence Quantum Optics Introduction For any vectors $\left\langle x_{1},x_{2}\right\rangle $ of $\mathbb{R}^{2}$ and $\left\langle y_{1},y_{2},y_{3}\right\rangle $ of $\mathbb{R}^{3}$, a product of these two vectors, which is denoted as $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, is defined as the matrix \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} & x_{2}y_{2} & x_{2}y_{3} \end{array}\right]. \] The collection of all the products $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, up to some equivalence, is called the tensor product of $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ and is denoted by $\mathbb{R}^{2}\otimes\mathbb{R}^{3}$. General Definition For any vector spaces $U,V,W$ over…