Dual Vector Space
The space of all linear functionals f:V→R, noted as V∗ The dual space has the same dimension as the corresponding vector space or, given a space V, with bases (v1,…,vn), there exists a dual space V∗ with a dual basis (v∗1,…,v∗n).
Math
Tensors
As a linear representation # A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector (2x2 matrix), etc. ...
Kronecker Delta
δij={0if i ≠ j 1if i = j
Bases
a basis for an n-dimensional vector space V is any ordered set of linearly independent vectors (e1,e2,…,en) An arbitrary vector x in V can be expressed as a linear combination of the basis vectors: x=n∑i=1eixi
See Bases Transformation, Coordinate Transformation
Orthonormal Basis
An orthonormal basis is a basis where all the vectors are one unit long and all perpendicular to each other (e.g. the Cartesian plane)
Bases Transformation
Consider two bases (e1,e2) and (˜e1,˜e2), where we consider the former the old basis and the latter the new basis. Each vector (˜e1,˜e2) can be expressed as a linear combination of (e1,e2): ˜e1=e1S11+e2S21 ˜e2=e1S12+e2S22
(1.0) is the basis transformation formula, and the object S is the direct transformation {Sji,1≤i,j≤2}, (assuming a 2x2 matrix) which can also be written in matrix form:
\begin{equation} \begin{bmatrix} \mathbf{\tilde{e}}_{1} & \mathbf{\tilde{e}}_{2} \end{bmatrix}\,=\, \begin{bmatrix} \mathbf{e}_{1} & \mathbf{e}_{2}\, \end{bmatrix} \begin{bmatrix} S^{1}_{1} & S^{1}_{2}\\\
...
Identity Matrix
I(X)=X
Where any nxn matrix is established via the Kronecker Delta, e.g.
Iij=δij
Invertible Matrix
A matrix, which when multiplied by another matrix, results in the identity matrix. AA−1=I
e.g.
[ab cd][d−b −ca]=[10 01]
Vector Space
also known as a linear space A collection of objects known as “vectors”. In the Euclidean space these can be visualized as simple arrows with a direction and a length, but this analogy will not necessarily translate to all spaces. Addition and multiplication of these objects (vectors) must adhere to a set of axioms for the set to be considered a “vector space”. Addition (+) +:V×V⟶V
...