Linear-Algebra

Bases Transformation

Linear-Algebra Math

Consider two bases \((\mathbf{e}_{1},\mathbf{e}_{2})\) and \((\mathbf{\tilde{e}}_{1},\mathbf{\tilde{e}}_{2})\), where we consider the former the old basis and the latter the new basis. Each vector \((\mathbf{\tilde{e}}_{1},\mathbf{\tilde{e}}_{2})\) can be expressed as a linear combination of \((\mathbf{e}_{1},\mathbf{e}_{2})\): \begin{equation} \mathbf{\tilde{e}}_{1}\,=\,\mathbf{e}_{1}S^{1}_{1}\,+\,\mathbf{e}_{2}S^{2}_{1}\\\ \tag{1.0}\\\ \mathbf{\tilde{e}}_{2}\,=\,\mathbf{e}_{1}S^{1}_{2}\,+\,\mathbf{e}_{2}S^{2}_{2} \end{equation} (1.0) is the basis transformation formula, and the object \(S\) is the direct transformation \(\{S^{j}_{i},\,1\,\leq\,i,\,j\,\leq\,2\}\), (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

Linear-Algebra Math

\begin{equation} \mathbf{I}(\mathbf{X})=\mathbf{X} \end{equation} Where any \(nxn\) matrix is established via the Kronecker Delta, e.g. \begin{equation} \mathbf{I}_{ij}\,=\,\delta_{ij} \end{equation}

Invertible Matrix

Linear-Algebra Math

A matrix, which when multiplied by another matrix, results in the identity matrix. \begin{equation} \mathbf{A}\mathbf{A}^{-1} = I \end{equation} e.g. \begin{equation} \begin{bmatrix} a & b\\\ c & d \end{bmatrix} \begin{bmatrix} d & -b\\\ -c & a \end{bmatrix}= \begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix} \end{equation}

Vector Space

Linear-Algebra Math

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 (+) \begin{equation} +\,:\,V\,\times\,V\,\longrightarrow\,V \end{equation} ...

Axioms (Vector Space)

Linear-Algebra Math

To qualify as a vector space, a set \(V\) and its associated operations of addition (\(+\)) and multiplication/scaling (\(\cdot\)) must adhere to the below: Associativity # \begin{equation} \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \end{equation} Commutivity # \begin{equation} \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \end{equation} Identity of Addition # There exists and element \(\mathbf{0}\,\in\,V\), called the zero vector, such that \(\mathbf{v} + \mathbf{0} = \mathbf{v}\) for all \(\mathbf{v}\,\in\,V\). ...