Uniform Distribution

Math Probability Cryptography

\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 assigns an equal probability to each element of a given set.

Gradient Descent

Machine-Learning Math Algorithms

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. the cost function) To find the minimum of the cost function, we take its derivative and “move along” the tangential line of steepest (negative) descent. Each “step” is determined by the coefficient \(\alpha\), which is called the Learning Rate. \begin{equation} \Theta_{j_{new}} := \Theta_{j_{old}} - \alpha\frac{\partial}{\partial\Theta_{j}}J(\Theta_{0},\Theta_{1}) \end{equation} ...


Linear-Algebra Math

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) \begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation} Simplistically, covectors can be thought of as “row vectors”, or: \begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation} This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. ...

Linear Algebra

Math Main Differential-Geometry

Bases Bases Transformation Coordinate Transformation Covectors Dual Space Identity Matrix Invertible Matrix Invertible Matrix Orthonormal Basis Tensor Product Tensors Vector Space Axioms (Vector Space)

Linear Mapping

Linear-Algebra Math

A mapping from \(\mathbf{V} \rightarrow \mathbf{W}\) that preserves the operations of addition and scalar multiplication. Also known as # Linear Map Linear Transformation Linear Function

Multilinear Map

Linear-Algebra Math

A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.

Dual Space

Linear-Algebra Math

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 or, given a space \(V\), with bases \((v_{1},…,v_{n})\), there exists a dual space \(V^{*}\) with a dual basis \((v^{*}_{1},…,v^{*}_{n})\).