[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:
- (associativity over addition)
- (commutativity over addition)
- There exists a zero vector for all in .
- Inverse elements over addition.
- Associativity of scalar multiplication.
- Scalar multiplication times identity.
- Distributivity of scalar over addition.
- Distributivity of two added scalars.
Common Vector Spaces
- Coordinate space
- 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.