Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools
Home › Linear Algebra
SECTIONLinear Algebra15 subsectionsBROWSE ALL ↓INTERACTIVE2 TOOLSLinear AlgebraVisual Tools14 toolsJump to ↓Linear AlgebraCalculators3 toolsJump to ↓fREFERENCE312 ITEMSfLinear AlgebraFormulas257 itemsJump to ↓AaLinear AlgebraTerms andDefinitions55 itemsJump to ↓§CORE TOPICS11 SUBSECTIONSMatrix Decompositions5 TOPICSJump to ↓Determinants4 TOPICSJump to ↓Eigenvalues and Eigenvectors4 TOPICSJump to ↓Systems of Linear Equations4 TOPICSJump to ↓Matrices in Linear Algebra5 TOPICSJump to ↓Orthogonality in LinearAlgebra5 TOPICSJump to ↓Linear Transformations5 TOPICSJump to ↓Vector Spaces7 TOPICSJump to ↓Vectors6 TOPICSJump to ↓Interactive ToolsJump to ↓Linear Algebra SymbolsReferenceJump to ↓

Introduction to Linear Algebra

Linear algebra is a field of mathematics that focuses on studying vectors, matrices, and the relationships between them, forming the mathematical framework for analyzing structures and transformations in multidimensional spaces. It introduces powerful tools to understand and solve problems where quantities interact linearly, making it fundamental to numerous disciplines.

This field begins with vectors—quantities that have both magnitude and direction—and their operations, such as addition and scaling. It extends to matrices, which are grid-like arrangements of numbers used to represent systems of equations or transformations. Learning how to manipulate matrices and understand their properties is a key part of linear algebra.

Students also explore vector spaces, the environments in which vectors live, and subspaces, which reveal structure and constraints within these spaces. Concepts like linear independence, span, and basis give insight into how vectors relate and interact. The study of linear transformations, which describe how vectors change under operations like rotations or scaling, helps build a deeper understanding of the subject.

To help students navigate these foundational concepts, we created a dedicated Matrix Theory section that provides an in-depth exploration of matrices - from basic definitions and notations to various matrix types and properties. This interactive guide covers matrix structure, indexing, special cases of square matrices, and key matrix properties, with clear mathematical notation and visual examples throughout. The section serves as both a comprehensive learning resource and a quick reference for understanding these essential building blocks of linear algebra.

Eigenvalues and eigenvectors, pivotal concepts in linear algebra, allow students to uncover hidden properties of transformations. Techniques like solving systems of equations, matrix decomposition, and understanding projections or orthogonality are practical outcomes of this study.

Ultimately, linear algebra provides a foundation for solving abstract and applied problems, developing skills to think logically, recognize patterns, and simplify complex systems. It equips students with a versatile toolkit for further studies in mathematics, sciences, engineering, and beyond.

Linear Algebra Visual Tools

14 toolsExplore Linear Algebra Visual Tools →
Each tool turns a piece of linear algebra into something you can manipulate, watch, and step through — matrix operations, row reduction, vector geometry, systems of equations, and more.
×=row × column
Explore Linear Algebra Visual Tools →

Linear Algebra Formulas

257 itemsSee All Linear Algebra Formulas →
Navigate through an essential collection of linear algebra formulas that power mathematical analysis and transformations. This guide presents key formulas across vector operations, matrix calculations, eigenvalues, and linear transformations — each equipped with clear notation, detailed explanations, and practical examples. You will find precise mathematical representations, component breakdowns, and specific use cases for over 15 fundamental formula categories. The organized structure helps you quickly locate and understand the tools you need, whether for solving equations, analyzing transformations, or applying linear algebra concepts in real-world scenarios. Perfect for students needing formula clarification, researchers requiring quick mathematical reference, or practitioners applying linear algebra in their work.
View All Linear Algebra Formulas →

Linear Algebra Terms and Definitions

55 itemsSee All Linear Algebra Terms and Definitions →
Discover essential linear algebra definitions that form the mathematical foundation for understanding vectors, matrices, and their relationships. This guide breaks down key terms from basic vector concepts like magnitude and direction to advanced matrix classifications and properties. Each definition includes precise mathematical notation, clear explanations, and visual examples to help grasp the concept. Whether you’re learning about vector spaces, exploring matrix types like triangular and symmetric matrices, or studying transformations, this organized reference makes complex linear algebra terminology accessible. The page serves as both a learning tool and a quick reference for students and practitioners, featuring interactive mathematical notation and practical examples throughout.
Vectors7
VectorAn ordered list of nn real numbers: v=(v1,v2,…,vn)∈Rn\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^nRead more →ScalarAn element of the underlying field — in standard linear algebra, a real number c∈Rc \in \mathbb{R}Read more →Magnitude (Norm)The length of a vector, measured as its distance from the origin:
∥v∥=v12+v22+⋯+vn2\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}
Read more →Unit VectorA vector u^\hat{\mathbf{u}} with ∥u^∥=1\|\hat{\mathbf{u}}\| = 1Read more →Dot ProductAn operation that takes two vectors and returns a scalar, computed by summing the products of corresponding components:
$$\mathbf{u} \cdot \mathbf{v}…Read more →Cross ProductA binary operation on two vectors in R3\mathbb{R}^3 that produces a vector perpendicular to both inputs:
$$\mathbf{u} \times \mathbf{v} =…Read more →Linear CombinationA sum of vectors, each multiplied by a scalar coefficient:
c1v1+c2v2+⋯+ckvkc_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k
Read more →
Vector SpaceA set VV equipped with vector addition and scalar multiplication satisfying the vector space axiomsRead more →SubspaceA nonempty subset W⊆VW \subseteq V that is itself a vector space under the same operationsRead more →SpanThe set of all linear combinations of a given collection of vectors:
$$\text{Span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots…Read more →Linear IndependenceVectors v1,…,vk\mathbf{v}_1, \ldots, \mathbf{v}_k are linearly independent if the only solution to
$$c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k =…Read more →BasisA set {v1,…,vn}\{\mathbf{v}_1, \ldots, \mathbf{v}_n\} that is linearly independent and spans the entire vector spaceRead more →DimensionThe number of vectors in any basis of a vector space VV, denoted dim⁡(V)\dim(V)Read more →Column SpaceThe set of all vectors expressible as AxA\mathbf{x} — equivalently, the span of the columns of AA:
$$\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x}…Read more →Null Space (Kernel)The set of all solutions to the homogeneous system Ax=0A\mathbf{x} = \mathbf{0}:
$$\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} =…Read more →Row SpaceThe span of the rows of a matrix, equivalently the column space of its transpose:
Row(A)=Col(AT)\text{Row}(A) = \text{Col}(A^T)
Read more →Left Null SpaceThe null space of the transpose ATA^T — the set of all vectors y\mathbf{y} satisfying ATy=0A^T\mathbf{y} = \mathbf{0}:
$$\text{Nul}(A^T) = \{\mathbf{y}…Read more →
MatrixA rectangular array of numbers with mm rows and nn columns: A∈Rm×nA \in \mathbb{R}^{m \times n}Read more →Square MatrixA matrix with equal numbers of rows and columns: A∈Rn×nA \in \mathbb{R}^{n \times n}Read more →Identity MatrixThe square matrix with 11s on the main diagonal and 00s elsewhere, denoted InI_n:
$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0…Read more →Symmetric MatrixA square matrix satisfying A=ATA = A^TRead more →Inverse MatrixA square matrix A−1A^{-1} such that AA−1=A−1A=IAA^{-1} = A^{-1}A = IRead more →Singular MatrixA square matrix AA with det⁡(A)=0\det(A) = 0Read more →RankThe number of linearly independent columns (equivalently, rows) in a matrix:
rank(A)=dim⁡(Col(A))=dim⁡(Row(A))\text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A))
Read more →TraceThe sum of the main diagonal entries of a square matrix:
tr(A)=a11+a22+⋯+ann=∑i=1naii\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}
Read more →Diagonal MatrixA square matrix where aij=0a_{ij} = 0 for all i≠ji \neq jRead more →Positive Definite MatrixA symmetric matrix AA satisfying xTAx>0\mathbf{x}^T A \mathbf{x} > 0 for all nonzero x\mathbf{x}Read more →
System of Linear EquationsA collection of equations Ax=bA\mathbf{x} = \mathbf{b} where AA is an m×nm \times n matrix and b∈Rm\mathbf{b} \in \mathbb{R}^mRead more →Augmented MatrixThe matrix formed by appending the right-hand side vector b\mathbf{b} as an additional column to the coefficient matrix AA, written…Read more →Row Echelon FormA matrix where:
• all zero rows are at the bottom
• each leading entry (pivot) is to the right of the pivot in the row above
• all entries below each…Read more →Reduced Row Echelon FormRow echelon form with the additional requirements:
• every pivot is 11
• each pivot is the only nonzero entry in its columnRead more →PivotThe first nonzero entry in each row of a matrix in row echelon formRead more →Homogeneous SystemA system of linear equations in which every equation equals zero: Ax=0A\mathbf{x} = \mathbf{0}Read more →
Linear TransformationA function T:V→WT: V \to W between vector spaces that preserves addition and scalar multiplication:
$$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) +…Read more →Image (Range)The set of all output vectors of a linear transformation:
Im(T)={T(v)∣v∈V}\text{Im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}
Read more →Matrix RepresentationA matrix AA such that T(v)=A[v]BT(\mathbf{v}) = A[\mathbf{v}]_{\mathcal{B}} for a chosen basis B\mathcal{B}Read more →Change of Basis MatrixA matrix PP that converts coordinates from one basis to another: [v]B′=P−1[v]B[\mathbf{v}]_{\mathcal{B}'} = P^{-1}[\mathbf{v}]_{\mathcal{B}}Read more →Similar MatricesMatrices AA and BB are similar if B=P−1APB = P^{-1}AP for some invertible matrix PPRead more →
EigenvalueA scalar λ\lambda such that Av=λvA\mathbf{v} = \lambda\mathbf{v} for some nonzero vector v\mathbf{v}Read more →EigenvectorA nonzero vector v\mathbf{v} such that Av=λvA\mathbf{v} = \lambda\mathbf{v} for some scalar λ\lambdaRead more →EigenspaceThe set of all eigenvectors for a given eigenvalue λ\lambda, together with the zero vector — equivalently, the null space of $(A - \lambda…Read more →Characteristic PolynomialThe polynomial whose roots are the eigenvalues of AA, obtained by computing:
p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I)
Read more →Algebraic MultiplicityThe multiplicity of λ\lambda as a root of the characteristic polynomialRead more →Geometric MultiplicityThe dimension of the eigenspace associated with an eigenvalue λ\lambda:
$$\text{geo. mult.}(\lambda) = \dim(E_\lambda) = \dim(\text{Nul}(A - \lambda…Read more →Singular ValueA nonnegative scalar measuring how much a matrix stretches space along each principal direction, derived from the eigenvalues of ATAA^TA:
$$\sigma_i =…Read more →
Inner ProductA function ⟨⋅,⋅⟩:V×V→R\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R} satisfying symmetry, linearity, and positive-definitenessRead more →Orthogonal VectorsVectors u\mathbf{u} and v\mathbf{v} are orthogonal if ⟨u,v⟩=0\langle \mathbf{u}, \mathbf{v} \rangle = 0Read more →Orthogonal SetA set of vectors {v1,…,vk}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} where ⟨vi,vj⟩=0\langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0 for all i≠ji \neq jRead more →Orthonormal SetAn orthogonal set where every vector is a unit vector: ⟨vi,vj⟩=δij\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}Read more →Orthogonal ComplementThe set of all vectors in VV that are orthogonal to every vector in a subspace WW:
$$W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v},…Read more →Orthogonal MatrixA square matrix QQ satisfying QTQ=QQT=IQ^TQ = QQ^T = IRead more →
View All Linear Algebra Terms and Definitions →

Matrix Decompositions

Explore Matrix Decompositions →
Overview of matrix decompositions: LU, QR, Cholesky, spectral, and SVD. When to use each factorization, computational costs, relationships between decompositions, and applications.
Explore Matrix Decompositions →

Determinants

Explore Determinants →
Learn matrix determinants: 2×2 and 3×3 formulas, cofactor expansion, Sarrus rule, row operations, singular vs invertible matrices, geometric interpretation as volume and orientation, and applications.
Explore Determinants →

Eigenvalues and Eigenvectors

Explore Eigenvalues and Eigenvectors →
Learn eigenvalues and eigenvectors: definition, characteristic equation, eigenspaces, geometric meaning, worked examples, trace/determinant relationships, and applications.
Explore Eigenvalues and Eigenvectors →

Systems of Linear Equations

Explore Systems of Linear Equations →
Systems of linear equations: matrix form Ax = b, augmented matrix, three possible outcomes, geometric interpretation, row operations, echelon form, homogeneous systems, and solvability conditions.
Explore Systems of Linear Equations →

Matrices in Linear Algebra

Explore Matrices in Linear Algebra →
Explore matrices in linear algebra through our detailed guide. Starting with matrix definitions and notations, the page explains matrix structure, elements, and indexing. You will learn to distinguish between different matrix types — from basic row and column matrices to more complex square matrices. The guide also covers essential matrix properties and dives into special cases of square matrices like diagonal and triangular forms. Each topic features clear mathematical notation and visual examples to reinforce your understanding of these fundamental concepts.
Inside the Matrix Guide
Definitions and Notations — how matrices are written using different types of brackets (square, parentheses, vertical bars) and basic matrix notation conventions
Elements, Structure and Indexing — how matrix elements are organized in rows and columns, the 1-based indexing system, and referencing specific elements by row and column indices
Types of Matrices — classifications based on shape (column, row, rectangular, square) and content type (numeric, variable/symbolic, mixed, zero)
Matrix Properties — size/dimension, rank, determinant, eigenvalues/eigenvectors, and trace, and their importance in matrix operations and transformations
Square Matrices and Special Cases — special diagonal patterns (diagonal, upper and lower triangular) and element relationships (symmetric, skew-symmetric, identity, scalar)
Explore Matrices in Linear Algebra →

Orthogonality in Linear Algebra

Explore Orthogonality in Linear Algebra →
Learn orthogonality in linear algebra: orthogonal vectors, complements, projections, inner products, orthonormal bases, Gram-Schmidt process, and least-squares approximation.
Explore Orthogonality in Linear Algebra →

Linear Transformations

Explore Linear Transformations →
Linear transformations: definition, examples in Rⁿ and abstract spaces, non-examples, basis determination, matrix connection, properties, composition, and geometric interpretation.
Explore Linear Transformations →

Vector Spaces

Explore Vector Spaces →
Vector spaces in linear algebra: ten axioms, Rⁿ and abstract examples, subspaces, linear independence, span, basis, dimension, and the four fundamental subspaces of a matrix.
Explore Vector Spaces →

Vectors

Explore Vectors →
Learn vectors in linear algebra: definition, notation, types (unit, zero, position), operations (addition, scalar multiplication), magnitude, dot product, cross product, and linear combinations.
Explore Vectors →

Linear Algebra Calculators

3 toolsExplore Linear Algebra Calculators →
Step-by-step calculators for the core computations of linear algebra: solve systems of linear equations, carry out matrix operations, and work with vectors — every calculator shows its full working, one operation at a time.
Explore Linear Algebra Calculators →

Interactive Tools

Calculator
Determinant Visual Calculator with Steps
Visual, interactive determinant calculator with step-by-step explanations.
Open tool → Visual Tool
Gaussian Elimination Visualizer
Learn the principles of Gaussian elimination with our interactive visual tool.
Open tool → Visual Tool
Matrix Multiplication Animated
Learn the process of matrix multiplication explained step after step with our interactive widget.
Open tool → Visual Tool
Matrix Transposition Visualizer
Learn the principles of matrix transposition using this interactive visual tool.
Open tool → Visual Tool
Matrix Types Generator
Learn different types of square matrices and their special features with this interactive visual tool.
Open tool →

Linear Algebra Symbols Reference

View Linear Algebra Symbols →

Our Linear Algebra Symbols page presents a well-organized collection of notation fundamental to matrix theory and vector spaces. This reference serves as a valuable resource for students and practitioners working with linear systems.

The page features symbols categorized by their mathematical applications, including matrix operations (A⊤, det(A), tr(A)), vector spaces (ℝⁿ, ⟨v,w⟩, ∥v∥), and eigenvalue concepts (Av = λv). It covers advanced topics like matrix decompositions (LU, QR, SVD) and linear transformations, alongside practical notation for representing matrices and vectors in LaTeX.

Every symbol includes its proper mathematical notation, corresponding LaTeX code for typesetting, and a brief explanation of its mathematical significance — making this an indispensable reference for anyone working with linear algebra in academic research or applications.

View Linear Algebra Symbols →