Math
Uniform Distribution
\begin{equation} x \in U:P(x) = \frac{1}{|U|} \end{equation} Where \(|U|\) is the size of the universe (set). In Probability Theory, a uniform distribution …
Read →Point Distribution
x0: P(x) = 1,∀ x ≠ x0: P(x) = 0 In Probability Theory, a point distribution is a distribution which assigns all the probability to a given point (set element).
Read →Gradient Descent
An optimization algorithm for finding the local minimum of a differentiable function. (The red arrows show the minimums of \(J(\Theta_{0},\Theta_{1})\), i.e. …
Read →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 Algebra
Bases Bases Transformation Coordinate Transformation Covectors Dual Space Identity Matrix Invertible Matrix Invertible Matrix Orthonormal Basis 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 →Kronecker Delta
\begin{equation} \delta_{ij}= \begin{cases} 0 & \text{if i \(\neq\) j}\\\ 1 & \text{if i \(=\) j} \end{cases} \end{equation}
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: …
Read →Invariance (Mathematics)
An property (of a mathematical object) is invariant if, after some operation(s) are applied, that property remains unchanged. For instance, in a geometrical …
Read →Manifold
A manifold is a topological space that locally resembles Euclidean space near each point. As defined in Physics An N-dimensional manifold of points is one for …
Read →Symmetric Relation
This definition is more clearly described sybollically: \begin{equation} \forall a,b \in X(aRb\,\iff\,bRa) \end{equation} Tao’s definition: Given any two …
Read →Transitive Relation
A relation \(R\) over a set \(X\) is transitive if for all elements x, y, z in \(X\), whenever \(R\) relates \(x\) to \(y\) and \(y\) to \(z\), then \(R\) also …
Read →Reflexive Relation
A binary relation \(R\) over a set \(X\) is reflexive if it relates every \(x \in X\) to itself. \begin{equation} \forall x \in X \,|\, xRx \end{equation} An …
Read →Set Theory
Concepts Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function
Read →Math
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts Bijective Function Injective Function Surjective Function Transitive …
Read →Triangle Inequality
For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. \(z \leq x + y\)
Read →Topology
Concepts Metric Space Open Balls Neighborhood Manifold Topological Space
Read →Topological Space
An ordered pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a collection of subsets of \(X\) satisfying: The empty set (\(\emptyset\)) and \(X\) belong …
Read →Surjective Function
A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).
Read →Statistics
Concepts Null Hypothesis
Read →Relations (Sets)
Definition A relation \(R\) from the elements of set \(A\) to the elements of set \(B\) is a subset of \(A \times B\). …alternatively… Let \(A\) …
Read →Real Analysis
The theoretical foundation which underlies calculus Peano Axioms Surjective Function Bijective Function Injective Function Cartesian Product
Read →Peano Axioms
A natural number is any element of the set \(\mathbb{N} = \{0, 1, 2, 3…\}\) I 0 is a natural number. II If \(n\) is a natural number, then n++ is also a …
Read →Open Balls
Let \((X, d)\) be a metric space. Let \(a \in X\) and \(\delta > 0\). The subset of \(X\) consisting of those points \(x \in X\) such that \(d(a, x) < …
Read →Neighborhood
The set of points surrounding a point \(p\) in a set \(V\), such that \(p \in \mathbb{R}^{n}\), within a radius \(\epsilon > 0\) See also: Open Balls
Read →Metric Space
A set and a function which defines the distance between two points on that set An ordered pair \((X, d)\), where \(X\) is an arbitrary set and \(d\) is a metric …
Read →Locality (Math)
Locality refers to: A property \(P\), of a point \(x\), which holds true near every point around \(x\). As an example: A sphere (and, more generally, a …
Read →Injective Function
A function is injective (“one-to-one”) if \(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\) i.e. Given a set \(X\) and a set \(Y\), no …
Read →Cartesian Product
If \(X\) and \(Y\) are sets, then we define the Cartesian Product \(X \times Y\) to be the collection of ordered pairs, e.g. \((x, y)\), whose first component …
Read →Cardinality
Equality in Cardinality Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).
Read →Binary Relation
A binary relation \(R\) over sets \(X\) and \(Y\) is a subset of the cartesian product of \(X \times Y\). Where \(X\) is the domain, or set of departure of …
Read →Bijective Function
A function is bijective or “invertible” if it is Both one-to-one and onto (injective and surjective) In other words: each element of one set is …
Read →Set Theory
Concepts Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function
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