[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. (associativity over addition)
  2. (commutativity over addition)
  3. There exists a zero vector for all in .
  4. Inverse elements over addition.
  5. Associativity of scalar multiplication.
  6. Scalar multiplication times identity.
  7. Distributivity of scalar over addition.
  8. Distributivity of two added scalars.

Common Vector Spaces

  • Coordinate space
  • 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.