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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

 

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.

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\langl...

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