# 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.

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.

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.

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
• A (0, 1)-tensor
• A (1, 1)-tensor
• A (2, 0)-tensor
• A (2, 1)-tensor
• A (2, 2)-tensor

in

• Index notation
• Bra-ket notation
• Array notation

For each array in the array notation, assume we’re working in 3D space.