What is the difference between different vector spaces?

Question

What is the difference between 2 vector spaces, V and W, when a transformation(linear map) is given as

T: V->W

is it just the difference between dimensionality of the 2 spaces?

If stress tensor is viewed as a transformation(linear map), are the area vector and traction vector in different vector spaces?

Answer 1

A linear transformation can be defined from $T: V\to W$ without any restriction on the dimension of the two vector spaces $V$ and $W$.

The stress tensor is a map

$$T: \mathbb{R^3}\to \mathbb{R^3}, \,\vec n\to \vec t_n$$

which relates the a unit-length direction vector $\vec n$ to the stress vector $\vec t_n$ at a point across a unitary surface perpendicular to $\vec n$.