What's the difference between a 2D vector and a matrix and a rank 2 tensor?

Question

I realize that maybe this is a super basic question for this website but I'm just an artist trying to teach themselves more math out of personal interest and I don't really understand how or why these things are divided up as they are.

I understand that an array is different than a vector in that a vector has magnitude.

And I understand that a matrix is a 2d array.

But vectors can be multiplied against vectors right? So they must be kinda in the same ballpark to be compatible?

And further confusing is that scalar and vectors are both ranks of tensors. Tensors seem to make sense but is a rank 2 tensor the same as a 2d vector? Wikipedia calls a rank 2 vector a dyadic. Dyadic has a different wiki article thank 2d vector so it must be different then right?

Answer 1

So I figured it out!

The confusing part is that the dimensions (AKA the rank AKA the number of indices) of a tensor is not the same as the spatial dimensions that might be used to describe it.

A 2D vector can be described using a 1D tensor: (x, y).

A 3D vector can be described using a 1D tensor: (x, y, z).

But a 2D tensor, aka a matrix is more like: ((x, y, z), (x, y, z), (x, y, z)) Or in proper math notation:

$\begin{bmatrix}x & y & z\\x & y & z\\ x & y & z\end{bmatrix}$

the matrix has 2 dimensions (it's a rectangle) but that's not the same thing as the dimensions of space it may or may not describe.

Whew this was good to figure out finally!

Mathematics needs to improve it's terminology to have less words used in multiple ways in different contexts. Maybe that's why tensor rank was adopted I guess.