Mathematics,  Premium

The Tensor Product: from vector spaces to categories

General Definition
The tensor product is not the same as the Cartesian product

A jump to categories


Artificial Intelligence
Quantum Optics


SUMMARY: We show why the tensor product is not the same as the Cartesian product, and we extend that result to categories. We then mention the use of the tensor product in applications, such as artificial intelligence and quantum optics.

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
x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\
x_{2}y_{1} & x_{2}y_{2} & x_{2}y_{3}
The collection of all the products $\left\langl...

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