Multidimensional Algebra

notation	meaning	basis	Alternative
	direct sum or Kronecker sum depending on context
	direct product or Kronecker product depending on context
^	exterior algebra
^k	exterior algebra kth blade
	scalar or real depending on context (see below)	e0
²	2 dimensional vector	e1,e2
³	3 dimensional vector	e1,e2,e3
n	n dimensional vector	e1…en
^²²	bivector based on 2D algebra	e12
^²³	bivector based on 3D algebra	e12,e31,e23
C	complex numbers
H	quaternions		³
O	octonions		7
aA	a is an element of the set A

Scalar

I have used the term 'scalar' interchangeably with the term 'real', that is, a continuous value that can be represented by a single number.

Strictly speaking the term 'scalar' should be reserved for a quantity that is used to scale a vector, that is change its magnitude without changing its direction, or in other words a scalar is the ratio of parallel vectors.

For instance I should not really call energy a scalar because there are no vectors involved.

I apologise for my lack of mathematical rigor here, its just that the word scalar seems to better express that it is not a vector and its less likely to cause confusion with the real part of a complex number. Also this (mis?)usage is quite common in the computer world.

Matrix

notation	meaning
[M]	matrix
[M]t	transpose of matrix (swap rows & columns)
[M]-1	inverse of matrix
[I]	identity matrix (ones on leading diagonal, otherwise zeros)

The individual elements of the matrix are numbered as follows,

m00	m01	m02	…	m0n
m10	m11	m12	…	m1n
m20	m21	m22	…	m2n
mp0	mp1	mp2	…	mpn

The first subscript represents the row, the second subscript represents the column

Tensor Notation

notation	example	meaning
subscript	e1	coordinate basis
superscript	x1	coordinate value
Einstein Summation Convention	eixi=e1x1+e2x2+e3x3=∑eixi	When the same index appears twice in an expression, once raised and once lowered, a sum is implied.
partial derivatives	∂a=∂/∂a

Tensors are discussed here.