[draft in progress] So what is a tensor? The simple, non-technical definition is that it is just a multi-dimensional array. It can have 0 dimensions and just be a number (scalar), it can be a euclidean vector, a matrix or a multivector.

A tensor defines a vector space. Vector spaces, in turn, define addition and scalar multiplication. The key is that they don’t move the origin and result in a regularly spaced, equal grid. (There isn’t any bending or folding of space.)

Eight axioms that vector spaces follow:

1. $u+(v+w) = (u+v)+w$ (associativity over addition)
2. $u+v = v+u$ (commutativity over addition)
3. There exists a zero vector $v + 0 = v$ for all $v$ in $V$.
4. $v+(-v)=0$ Inverse elements over addition.
5. $a(b v ) = (ab) v$ Associativity of scalar multiplication.
6. $1v = v$ Scalar multiplication times identity.
7. $a(u+v) = a u + a v$ Distributivity of scalar over addition.
8. $(a+b)v = a v + b v$ Distributivity of two added scalars.

### Common Vector Spaces

• Coordinate space $\Re^n$
• Matricies
• Polynomial Functions
• Complex numbers
• Continuous functions
• Differential equations

The determinant can be seen as a scale factor for transformation of area. A determinant of zero means that the transformed space collapsed into a smaller subspace.