Definitions, Symbols, Formulas, and Notes — All in One Place.
Show Shortcuts
Vectors
Term | Definition | Symbol | |||
|---|---|---|---|---|---|
| Vector | An ordered set of numbers or symbols, typically arranged in a column or row. | \(\vec{v}\) | |||
| Unit Vector | A vector with a magnitude of 1. | \(\hat{v}\) | |||
| Scalar | A single, standalone numerical value, typically used to represent a quantity such as a magnitude. | \(\lambda\) | |||
| Vector Magnitude | The length of a vector. It may be calculated by squaring all elements in a vector, then calculating the square root of their sum. For instance, a two-dimensional vector, \(\vec{v}\)=\([x,y]\), \( |\vec{v}|=\sqrt{x^2 + y^2} \) | \(|\vec{v}|\) | |||
Matrices
Term | Definitions | Symbol | |||
|---|---|---|---|---|---|
| Matrix | A two-dimensional vector of numbers, symbols, or expressions arranged in rows and columns. | ||||
| Identity Matrix | A square matrix with ones on the main diagonal and zeros elsewhere. | \(I\) | |||
| Determinant | A scalar value that can be computed from the elements of a square matrix. | ||||
| Matrix Inverse | If a matrix has an inverse, multiplying the matrix by its inverse results in the identity matrix. | \(AA^{-1} = A^{-1}A = I\) | |||
| Pseudoinverse | A generalization of the matrix inverse when the matrix is not necessarily square or invertible. | \(A^+\) | |||
Matrix Types
Term | Definition | ||
|---|---|---|---|
| Square Matrix | A matrix with the same number of rows and columns. | ||
| Diagonal Matrix | A matrix where all elements outside the main diagonal are zero. | ||
| Symmetric Matrix | A square matrix that is equal to its transpose. | ||
| Dense Matrix | A matrix in which most of the elements are non-zero. | ||
| Sparse Matrix | A matrix in which most of the elements are zero. | ||
| Cofactor Matrix | A matrix in which each element is the cofactor of the corresponding element of the original matrix, often used in computing determinants and matrix inverses. | ||
| Coefficient Matrix | In the context of a system of linear equations, the matrix formed by the coefficients of the variables. | ||
| Hilbert Matrix | A square matrix whose entries are the reciprocals of the sums of their row and column indices plus one. | ||
| Sub Matrix | A matrix formed by selecting certain rows and columns from a larger matrix. | ||
| Singular Matrix | A square matrix that is not invertible, meaning its determinant is zero. | ||
| Markov Matrix | A square matrix used to describe transitions between states in a Markov chain, where each element represents the probability of transitioning from one state to another. | ||
| Consumption Matrix | In the context of economics or mathematical modeling, a matrix representing the consumption rates of different resources by different entities. | ||
| Lower Triangular Matrix | A square matrix in which all the entries above the main diagonal are zero. | ||
| Change of Basis Matrix | A matrix that represents the linear transformation between two different bases in a vector space. | ||
| Upper Triangular Matrix | A square matrix in which all the entries below the main diagonal are zero. | ||
Matrix Operations
Term | Definitions | Formula | |||
|---|---|---|---|---|---|
| Transpose | The result of swapping a matrix's rows with its columns. | \(A^T\) | |||
| Inner Product | A binary operation that takes two vectors and produces a scalar. | \(\vec{v} \cdot \vec{u}\) | |||
| Outer Product | A mathematical operation that takes two vectors as input and produces a matrix as output. | \(\vec{v} \otimes \vec{u}\) | |||
| Matrix Multiplication | An operation that takes two matrices and produces another matrix by multiplying corresponding entries and summing the results. | \(C = A \cdot B\) | |||
| Hadamard Product | An element-wise product of two matrices of the same dimensions. | ||||
| Trace | The sum of the diagonal elements of a square matrix. | ||||
| Norm | A measure of a matrix's size. | \(||\mathbf{v}||\) | |||
| Matrix Conjugate | The matrix that results from changing the sign of each imaginary number in the original matrix. | \( A^* \) | |||
| Singular Value | A singular value, denoted as \(\sigma_i\), is a scalar that represents the magnitude of stretching or scaling associated with a particular direction in the transformation defined by a matrix. | ||||
| Singular Value Decomposition | A factorization of a matrix into three other matrices, including a diagonal matrix of singular values. | ||||
Linear Equations
Term | Definitions | ||
|---|---|---|---|
| QR Decomposition | A decomposition of a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R). | ||
| LU Decomposition | A factorization of a square matrix into a product of a lower triangular matrix (L) and an upper triangular matrix (U). | ||
| Jordan Decomposition | A way of representing a square matrix as the sum of a diagonal matrix and a nilpotent matrix. | ||
| Free Variables | Variables in a linear system of equations that can take on any value, often corresponding to variables not leading to unique solutions. | ||
| Free Columns | Columns in a matrix that do not contain a pivot element when reduced to row-echelon form. | ||
| Pivot Variables | Variables in a linear system of equations corresponding to columns containing pivot elements. | ||
Row-Echelon Form | A matrix is in row-echelon form if: All zero rows, if any, are at the bottom of the matrix. The leading entry of each nonzero row is in a column to the right of the leading entry of the previous row. The leading entry in each nonzero row is 1. All entries in the column below a leading 1 are zero. | ||
| Reduced Row-Echelon Form | A matrix is in reduced row-echelon form if it satisfies the conditions of row-echelon form and the following additional conditions: The leading 1 in each nonzero row is the only nonzero entry in its column. All entries in the column above and below a leading 1 are zero. | ||
Subspaces
Term | Definitions | Symbol | |||
|---|---|---|---|---|---|
| Linear Independence | A set of vectors is linearly independent if no vector in the set is a linear combination of the others. | \(\text{det}(V) \neq 0\) | |||
| Vector Space | The set of all possible vectors satisfying certain properties. | ||||
| Row Space | The vector space spanned by the rows of a matrix. | \(\text{span}(A^T)\) | |||
| Column Space | The vector space spanned by the columns of a matrix. | \(\text{span}(A)\) | |||
| Nullspace | The set of all solutions to the homogeneous equation\( Ax = 0\), where \(A\) is a matrix. | ||||
| Rank | The maximum number of linearly independent rows or columns in a matrix. | ||||
| Linear Combination | A linear combination is an expression constructed from vectors by multiplying them by scalars and adding the results. | \(\sum_{i=1}^{n} c_i\vec{v}_i\) | |||
| Span | The set of all possible linear combinations of vectors in a vector space. | \(\text{span}(\vec{v}_1, \vec{v}_2, ..., \vec{v}_n)\) | |||
| Basis | A set of vectors that spans a vector space and is linearly independent. | \(\vec{v}_1, \vec{v}_2, ..., \vec{v}_n\) | |||
| Dimension | The number of vectors in a basis for a vector space. | \(\text{dim}(\mathbb{V})\) | |||
| Full Column Rank | A matrix whose columns are linearly independent, meaning the rank of the matrix is equal to the number of columns. | ||||
| Full Row Rank | A matrix whose rows are linearly independent, meaning the rank of the matrix is equal to the number of rows. | ||||
Orthogonality
Term | Definitions | Formula | |||
|---|---|---|---|---|---|
| Orthogonal Matrix | A square matrix whose rows and columns are perpendicular to each other. | \(Q^T = Q^{-1}\) | |||
| Orthonormal Matrix | A matrix where the vectors are not only orthogonal to each other but also normalized to have a length of 1. | \(Q^TQ = QQ^T = I\) | |||
| Projection | A geometric operation that involves finding the component of one vector along another. | \(\text{proj}_{\mathbb{U}}(\vec{v}) = \frac{\vec{v} \cdot \vec{u}}{\|\vec{u}\|^2}\vec{u}\) | |||
| Change of Basis | The process of expressing vectors in one basis using coordinates from another basis. | \(P^{-1}AP\) | |||
| Gram-Schmidt | A process that orthogonalizes a linearly independent set of vectors to form an orthonormal basis. | ||||
Eigenvalues and Eigenvectors
Term | Definitions | Formula | |||
|---|---|---|---|---|---|
| Eigenvalue | A scalar \(\lambda\) such that there exists a non-zero vector \(v\) for which \(Av = \lambda v\) | \(\text{det}(A - \lambda I) = 0\) | |||
| Eigenvector | A non-zero vector \(v\) that only gets scaled by a scalar (called the eigenvalue) when multiplied by a matrix A. | \((A - \lambda I)\vec{v} = 0\) | |||
| Principal Component | The direction in which the data in a matrix varies the most. It is the eigenvector corresponding to the largest eigenvalue of the covariance matrix of the data. | ||||
| Covariance Matrix | A matrix that summarizes covariance relationships between variables in a dataset. The diagonal elements represent variances, and off-diagonal elements represent covariances. | ||||
| Repeated Eigenvalues | Eigenvalues that occur more than once, indicating that the corresponding eigenvectors are not linearly independent. | ||||
| Sum of Eigenvalues | Equal to the trace of the matrix in a square matrix. | ||||
| Positive Definite Matrix | A symmetric matrix where all eigenvalues are positive. | ||||
Theorems
Term | Definitions | ||
|---|---|---|---|
| Rank-Nullity Theorem | A theorem stating that the dimension of the column space plus the dimension of the null space of a matrix equals the number of columns in the matrix. | ||
| Uniqueness of Reduced Row-Echelon Form | Every matrix \(A\) may be represented by a unique matrix in reduced row-echelon form. | ||
| Spectral Theorem | A theorem stating that states that every symmetric matrix is diagonalizable and its eigenvalues are real. | ||
Linear Transformations
Term | Definitions | ||
|---|---|---|---|
| Kernel | The set of all vectors that map to the zero vector in the codomain. | ||
Image Codomain | The set of all possible outputs that can be obtained by applying a transformation to the vectors in its domain. | ||
Onto Matrix Surjective Matrix | A matrix transformation that maps a higher-dimensional vector space onto a lower-dimensional one. Every element in the codomain is mapped to by at least one element from the domain. | ||
One-to-One Matrix Injective Matrix | A matrix transformation where distinct elements in the domain map to distinct elements in the codomain. Each element in the domain maps to a unique element in the codomain, and no two distinct elements in the domain map to the same element in the codomain. | ||
Applications
Term | Definition | ||
|---|---|---|---|
| Least Squares Regression | |||
| SVM Classification | |||
| SVM Dual | |||
| Logistic Regression | |||
To pick up a draggable item, press the space bar.
While dragging, use the arrow keys to move the item.
Press space again to drop the item in its new position, or press escape to cancel.
AI Study Tools for STEM Students Worldwide.
© 2025 CompSciLib™, LLC. All rights reserved.
info@compscilib.comContact Us