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}\\\
...I(X)=X
Where any nxn matrix is established via the Kronecker Delta, e.g.
Iij=δij
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]
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
...To qualify as a vector space, a set V and its associated operations of addition (+) and multiplication/scaling (⋅) must adhere to the below:
Associativity # u+(v+w)=(u+v)+w
Commutivity # u+v=v+u
Identity of Addition # There exists and element 0∈V, called the zero vector, such that v+0=v for all v∈V.
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