Linear-Algebra
Covectors
A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar) …
Read →Differential Geometry
Tensors Tensor Product
Read →Linear Mapping
A mapping from \(\mathbf{V} \rightarrow \mathbf{W}\) that preserves the operations of addition and scalar multiplication. Also known as Linear Map Linear …
Read →Multilinear Map
A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.
Read →Tensor Product
Read →Dual Space
The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\) The dual space has the same dimension as the corresponding vector space …
Read →Dual Vector Space
The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\) The dual space has the same dimension as the corresponding vector space …
Read →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 …
Read →Bases
a basis for an n-dimensional vector space \(V\) is any ordered set of linearly independent vectors \((\mathbf{e}_{1}, \mathbf{e}_{2},…,\mathbf{e}_{n})\) …
Read →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)
Read →Bases Transformation
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 …
Read →Identity Matrix
\begin{equation} \mathbf{I}(\mathbf{X})=\mathbf{X} \end{equation} Where any \(nxn\) matrix is established via the Kronecker Delta, e.g. \begin{equation} …
Read →Invertible Matrix
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. …
Read →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 …
Read →Coordinate Transformation
Read →Axioms (Vector Space)
To qualify as a vector space, a set \(V\) and its associated operations of addition (\(+\)) and multiplication/scaling (\(\cdot\)) must adhere to the below: …
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