A Mathematical Approach to Tensors

A matrix is a linear map of a vector to another vector. A tensor is a multilinear map of a tensor to another tensor.

This is part of a matrix. This is stop 20 on the Road to Quantum Mechanics.

There are a lot of bad explanations in science and math.

• “Spin is like a ball that’s spinning, except it’s not a ball and it’s not spinning.”
• “Entropy is disorder.”
• “A vector is an element of a vector space.”
• “A tensor is a thing that transforms like a tensor.”

The last two reflect a common mindset in STEM that you don’t need scaffolding to build a concept just because the final building has none.

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While I have some strong words about these explanations, they’re best suited for other articles and videos. In this article, we’re going to unify several mathematical threads that have been running throughout this series.

Check Your Understanding

You’re going to work with tensors both as arbitrary objects and in terms of specific examples.

Tensors in Different Notations

Write out the following tensors

• A (1, 0)-tensor