Linear Algebra Terms and Definitions

AAlgebraic MultiplicityEigen AAugmented MatrixLinear Systems BBasisVector Spaces CChange of Basis MatrixTransformations CCharacteristic PolynomialEigen CCofactorDeterminants CCofactor Matrix (Adjugate)Determinants CColumn SpaceVector Spaces CCross ProductVectors DDeterminantDeterminants DDiagonal MatrixMatrices DDimensionVector Spaces DDot ProductVectors EEigenspaceEigen EEigenvalueEigen EEigenvectorEigen GGeometric MultiplicityEigen HHomogeneous SystemLinear Systems IIdentity MatrixMatrices IImage (Range)Transformations IInner ProductOrthogonality IInverse MatrixMatrices LLeft Null SpaceVector Spaces LLinear CombinationVectors LLinear IndependenceVector Spaces LLinear TransformationTransformations MMagnitude (Norm)Vectors MMatrixMatrices MMatrix RepresentationTransformations MMinorDeterminants NNull Space (Kernel)Vector Spaces OOrthogonal ComplementOrthogonality OOrthogonal MatrixOrthogonality OOrthogonal SetOrthogonality OOrthogonal VectorsOrthogonality OOrthonormal SetOrthogonality PPivotLinear Systems PPositive Definite MatrixMatrices RRankMatrices RReduced Row Echelon FormLinear Systems RRow Echelon FormLinear Systems RRow SpaceVector Spaces SScalarVectors SSimilar MatricesTransformations SSingular MatrixMatrices SSingular ValueEigen SSpanVector Spaces SSquare MatrixMatrices SSubspaceVector Spaces SSymmetric MatrixMatrices SSystem of Linear EquationsLinear Systems TTraceMatrices UUnit VectorVectors VVectorVectors VVector SpaceVector Spaces

Determinants(4)

Eigen(7)

Linear Systems(6)

Matrices(10)

Orthogonality(6)

Transformations(5)

Vector Spaces(10)

Vectors(7)

55 of 55 terms

Determinants

Determinant Minor Cofactor Cofactor Matrix (Adjugate)

Eigen

Eigenvalue Eigenvector Eigenspace Characteristic Polynomial Algebraic Multiplicity Geometric Multiplicity Singular Value

Linear Systems

System of Linear Equations Augmented Matrix Row Echelon Form Reduced Row Echelon Form Pivot Homogeneous System

Matrices

Matrix Square Matrix Identity Matrix Symmetric Matrix Inverse Matrix Singular Matrix Rank Trace Diagonal Matrix Positive Definite Matrix

Orthogonality

Inner Product Orthogonal Vectors Orthogonal Set Orthonormal Set Orthogonal Complement Orthogonal Matrix

Transformations

Linear Transformation Image (Range)Matrix Representation Change of Basis Matrix Similar Matrices

Vector Spaces

Vector Space Subspace Span Linear Independence Basis Dimension Column Space Null Space (Kernel)Row Space Left Null Space

Vectors

Vector Scalar Magnitude (Norm)Unit Vector Dot Product Cross Product Linear Combination

55 terms

Vectors

(7 items)

Vector

An ordered list of

n

real numbers:

\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n

Vectors

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A quantity with both magnitude and direction. In

\mathbb{R}^2

and

\mathbb{R}^3

, vectors can be visualized as arrows from the origin to a point.

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Scalar

An element of the underlying field — in standard linear algebra, a real number

c \in \mathbb{R}

Vectors

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A single number used to scale vectors. Multiplying a vector by a scalar changes its length without altering its direction (unless the scalar is negative, which reverses direction).

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Magnitude (Norm)

The length of a vector, measured as its distance from the origin:

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Vector Magnitude

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\mathbb{R}^2

, this reduces to the hypotenuse given by the Pythagorean theorem. The concept generalizes to any

\mathbb{R}^n

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Unit Vector

A vector

\hat{\mathbf{u}}

with

\|\hat{\mathbf{u}}\| = 1

Vector Magnitude

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A vector that encodes direction only, with all length information removed. Any nonzero vector

\mathbf{v}

can be normalized to a unit vector by dividing by its magnitude:

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

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Dot Product

An operation that takes two vectors and returns a scalar, computed by summing the products of corresponding components:

\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = \|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta

Dot Product

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A scalar measure of how much two vectors point in the same direction. Positive when the angle between them is acute, zero when perpendicular, negative when obtuse.

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Cross Product

A binary operation on two vectors in

\mathbb{R}^3

that produces a vector perpendicular to both inputs:

\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}

Cross Product

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Its magnitude equals the area of the parallelogram spanned by the two vectors. The direction follows the right-hand rule.

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Linear Combination

A sum of vectors, each multiplied by a scalar coefficient:

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k

Linear Combinations

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A new vector built by scaling given vectors and adding the results. The set of all possible linear combinations of a collection of vectors defines their span.

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Vector Spaces

(10 items)

Vector Space

A set

V

equipped with vector addition and scalar multiplication satisfying the vector space axioms

Vector Spaces

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A collection of objects (vectors) that can be added together and scaled by numbers, where these operations behave predictably.

\mathbb{R}^n

is the most familiar example, but the concept extends to function spaces, polynomial spaces, and more.

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Subspace

A nonempty subset

W \subseteq V

that is itself a vector space under the same operations

Subspaces

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A subset of a vector space that is closed under addition and scalar multiplication. Every subspace must contain the zero vector.

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Span

The 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 + c_k\mathbf{v}_k \mid c_i \in \mathbb{R}\}

Span

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Geometrically, spanning two non-parallel vectors in

\mathbb{R}^3

gives a plane; spanning three linearly independent vectors fills the entire space.

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Linear Independence

Vectors

\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 = \mathbf{0}

c_1 = c_2 = \cdots = c_k = 0

Linear Independence

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No vector in the set can be written as a linear combination of the others. Each vector contributes a genuinely new direction. Removing any one reduces the span.

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Basis

A set

\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}

that is linearly independent and spans the entire vector space

Vector Spaces

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A minimal set of vectors that can produce every vector in the space through linear combinations. Every vector has a unique representation as a linear combination of basis vectors.

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Dimension

The number of vectors in any basis of a vector space

V

, denoted

\dim(V)

Dimension

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The number of independent directions in a space.

\mathbb{R}^3

is three-dimensional because any basis has exactly three vectors.

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Column Space

The set of all vectors expressible as

A\mathbf{x}

— equivalently, the span of the columns of

A

\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}

Fundamental Spaces

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The set of all vectors

\mathbf{b}

for which

A\mathbf{x} = \mathbf{b}

has a solution. It captures everything the matrix can "reach" as output.

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Null Space (Kernel)

The set of all solutions to the homogeneous system

A\mathbf{x} = \mathbf{0}

\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}

Fundamental Spaces

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The set of all inputs that the matrix sends to the zero vector. If the null space contains only

\mathbf{0}

, the matrix is injective (one-to-one). A larger null space means the matrix "collapses" some directions.

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Row Space

The span of the rows of a matrix, equivalently the column space of its transpose:

\text{Row}(A) = \text{Col}(A^T)

Fundamental Spaces

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A subspace of

\mathbb{R}^n

spanned by the rows of

A

. Row operations change individual rows but preserve the row space, making echelon form useful for finding a basis.

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Left Null Space

The null space of the transpose

A^T

— the set of all vectors

\mathbf{y}

satisfying

A^T\mathbf{y} = \mathbf{0}

\text{Nul}(A^T) = \{\mathbf{y} \in \mathbb{R}^m \mid A^T\mathbf{y} = \mathbf{0}\}

Fundamental Spaces

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It completes the four fundamental subspaces: column space, null space, row space, and left null space.

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Matrices

(10 items)

Matrix

A rectangular array of numbers with

m

rows and

n

columns:

A \in \mathbb{R}^{m \times n}

Matrices

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A compact way to encode a linear transformation or a system of linear equations. The entry in row

i

, column

j

is denoted

a_{ij}

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Square Matrix

A matrix with equal numbers of rows and columns:

A \in \mathbb{R}^{n \times n}

Matrix Types

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Only square matrices can have determinants, eigenvalues, inverses, and a trace. They represent transformations from a space to itself.

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Identity Matrix

The square matrix with

1

s on the main diagonal and

0

s elsewhere, denoted

I_n

I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}

Matrix Types

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The matrix that leaves every vector unchanged:

I\mathbf{v} = \mathbf{v}

. It is the multiplicative identity for matrices, analogous to the number

1

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Symmetric Matrix

A square matrix satisfying

A = A^T

Matrix Types

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A matrix that equals its own transpose — entries are mirrored across the main diagonal. Symmetric matrices arise naturally in distance, covariance, and quadratic form problems.

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Inverse Matrix

A square matrix

A^{-1}

such that

AA^{-1} = A^{-1}A = I

Matrix Inverse

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The matrix that "undoes" the transformation applied by

A

. If

A

maps

\mathbf{x}

\mathbf{b}

, then

A^{-1}

maps

\mathbf{b}

back to

\mathbf{x}

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Singular Matrix

A square matrix

A

with

\det(A) = 0

Determinants

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A matrix that collapses at least one dimension — it maps some nonzero vector to

\mathbf{0}

. A singular matrix has no inverse and the system

A\mathbf{x} = \mathbf{b}

either has no solution or infinitely many.

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Rank

The number of linearly independent columns (equivalently, rows) in a matrix:

\text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A))

Matrix Rank

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It measures the "effective dimensionality" of the transformation — how many independent output directions the matrix produces.

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Trace

The sum of the main diagonal entries of a square matrix:

\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}

Matrix Trace

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It equals the sum of all eigenvalues (counted with multiplicity), providing a quick invariant of the matrix.

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Diagonal Matrix

A square matrix where

a_{ij} = 0

for all

i \neq j

Matrix Types

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A matrix whose only nonzero entries lie on the main diagonal. Diagonal matrices scale each coordinate axis independently, making them the simplest matrices to work with.

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Positive Definite Matrix

A symmetric matrix

A

satisfying

\mathbf{x}^T A \mathbf{x} > 0

for all nonzero

\mathbf{x}

Cholesky Decomposition

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A matrix where the associated quadratic form is always positive — it curves upward in every direction, like a bowl. Positive definite matrices generalize the idea of a positive number to matrix algebra.

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Determinants

(4 items)

Determinant

A scalar

\det(A) \in \mathbb{R}

assigned to every square matrix, defined recursively via cofactor expansion

Determinants

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The signed volume scaling factor of the linear transformation represented by the matrix. A determinant of zero means the transformation collapses space into a lower dimension.

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Minor

The determinant of the submatrix obtained by deleting row

i

and column

j

from a matrix:

M_{ij} = \det(\hat{A}_{ij})

Cofactors

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Minors are the building blocks for cofactors and, through them, for the full determinant computation.

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Cofactor

A signed minor, with sign determined by the position

(i,j)

C_{ij} = (-1)^{i+j} M_{ij}

Cofactors

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The sign alternates in a checkerboard pattern (

+, -, +, \ldots

). Cofactors are used in the expansion formula for determinants and in computing the adjugate.

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Cofactor Matrix (Adjugate)

The transpose of the matrix of cofactors of

A

\text{adj}(A) = C^T

Cofactors

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It provides a formula for the inverse:

A^{-1} = \frac{1}{\det(A)}\,\text{adj}(A)

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Linear Systems

(6 items)

System of Linear Equations

A collection of equations

A\mathbf{x} = \mathbf{b}

where

A

is an

m \times n

matrix and

\mathbf{b} \in \mathbb{R}^m

Linear Systems

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A set of linear equations sharing the same unknowns. The system has either no solution, exactly one solution, or infinitely many solutions — there is no other possibility.

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Augmented Matrix

The matrix formed by appending the right-hand side vector

\mathbf{b}

as an additional column to the coefficient matrix

A

, written

[A \mid \mathbf{b}]

Linear Systems

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A compact notation that combines the coefficient matrix and the right-hand side into a single matrix, so row operations can be applied to the entire system at once.

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Row Echelon Form

A 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 pivot are zero

Echelon Form

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A staircase pattern achieved by row operations, making back-substitution possible. The number of pivots equals the rank.

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Reduced Row Echelon Form

Row echelon form with the additional requirements:
• every pivot is

1

• each pivot is the only nonzero entry in its column

Echelon Form

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The fully simplified form of a matrix under row operations. Unlike row echelon form, the reduced form is unique — every matrix has exactly one RREF.

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Pivot

The first nonzero entry in each row of a matrix in row echelon form

Echelon Form

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Pivots mark the "independent" columns. The number of pivots determines the rank; columns without pivots correspond to free variables in the solution.

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Homogeneous System

A system of linear equations in which every equation equals zero:

A\mathbf{x} = \mathbf{0}

Homogeneous Systems

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It always has at least the trivial solution

\mathbf{x} = \mathbf{0}

. Nontrivial solutions exist if and only if

\text{rank}(A) < n

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Transformations

(5 items)

Linear Transformation

A function

T: V \to W

between vector spaces that preserves addition and scalar multiplication:

T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})

T(c\mathbf{u}) = cT(\mathbf{u})

Linear Transformations

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Lines map to lines (or to the origin), and the origin stays fixed. Every linear transformation between finite-dimensional spaces can be represented by a matrix.

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Image (Range)

The set of all output vectors of a linear transformation:

\text{Im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}

Image and Kernel

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For a matrix transformation

T(\mathbf{x}) = A\mathbf{x}

, the image equals the column space of

A

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Matrix Representation

A matrix

A

such that

T(\mathbf{v}) = A[\mathbf{v}]_{\mathcal{B}}

for a chosen basis

\mathcal{B}

Matrix Representation

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Every linear transformation between finite-dimensional spaces can be encoded as a matrix once bases are chosen. Different bases produce different matrices representing the same transformation.

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Change of Basis Matrix

A matrix

P

that converts coordinates from one basis to another:

[\mathbf{v}]_{\mathcal{B}'} = P^{-1}[\mathbf{v}]_{\mathcal{B}}

Basis Change

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The same vector can be described using different coordinate systems (bases). The change of basis matrix translates between these coordinate systems.

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Similar Matrices

Matrices

A

and

B

are similar if

B = P^{-1}AP

for some invertible matrix

P

Basis Change

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Two matrices are similar when they represent the same linear transformation under different bases. They share all basis-independent properties.

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Eigen

(7 items)

Eigenvalue

A scalar

\lambda

such that

A\mathbf{v} = \lambda\mathbf{v}

for some nonzero vector

\mathbf{v}

Eigenvalues and Eigenvectors

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The factor by which the linear transformation stretches or compresses an eigenvector. A negative eigenvalue means the eigenvector's direction is reversed.

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Eigenvector

A nonzero vector

\mathbf{v}

such that

A\mathbf{v} = \lambda\mathbf{v}

for some scalar

\lambda

Eigenvalues and Eigenvectors

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A direction that the matrix preserves — the transformation only stretches or compresses along this direction without rotating it. Eigenvectors from distinct eigenvalues are always linearly independent.

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Eigenspace

The set of all eigenvectors for a given eigenvalue

\lambda

, together with the zero vector — equivalently, the null space of

(A - \lambda I)

E_\lambda = \text{Nul}(A - \lambda I)

Eigenvalues and Eigenvectors

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It is always a subspace, and its dimension reveals how many independent eigenvector directions exist for that eigenvalue.

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Characteristic Polynomial

The polynomial whose roots are the eigenvalues of

A

, obtained by computing:

p(\lambda) = \det(A - \lambda I)

Characteristic Equation

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For an

n \times n

matrix, the characteristic polynomial has degree

n

, so there are at most

n

eigenvalues (counted with multiplicity).

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Algebraic Multiplicity

The multiplicity of

\lambda

as a root of the characteristic polynomial

Characteristic Equation

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How many times the eigenvalue appears as a root. If

p(\lambda) = (\lambda - 2)^3(\lambda + 1)

, then

\lambda = 2

has algebraic multiplicity

3

and

\lambda = -1

has algebraic multiplicity

1

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Geometric Multiplicity

The dimension of the eigenspace associated with an eigenvalue

\lambda

\text{geo. mult.}(\lambda) = \dim(E_\lambda) = \dim(\text{Nul}(A - \lambda I))

Eigenvalue Properties

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The number of independent eigenvector directions for a given eigenvalue. A matrix is diagonalizable if and only if every eigenvalue has geometric multiplicity equal to its algebraic multiplicity.

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Singular Value

A nonnegative scalar measuring how much a matrix stretches space along each principal direction, derived from the eigenvalues of

A^TA

\sigma_i = \sqrt{\lambda_i(A^TA)}

SVD

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Unlike eigenvalues, singular values are always nonnegative and exist for any matrix, not just square ones.

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Orthogonality

(6 items)

Inner Product

A function

\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}

satisfying symmetry, linearity, and positive-definiteness

Inner Product

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A generalization of the dot product to abstract vector spaces. It defines notions of length, angle, and orthogonality in spaces where the standard dot product may not apply.

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Orthogonal Vectors

Vectors

\mathbf{u}

and

\mathbf{v}

are orthogonal if

\langle \mathbf{u}, \mathbf{v} \rangle = 0

Orthogonality

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Perpendicular vectors. Their dot product (or inner product) is zero, meaning they share no component in each other's direction.

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Orthogonal Set

A set of vectors

\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}

where

\langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0

for all

i \neq j

Orthogonal Sets

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A collection of mutually perpendicular vectors. Every orthogonal set of nonzero vectors is automatically linearly independent.

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Orthonormal Set

An orthogonal set where every vector is a unit vector:

\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}

Orthogonal Sets

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Mutually perpendicular vectors, each of length

1

. An orthonormal basis gives the simplest coordinate system — coordinates are just inner products with the basis vectors.

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Orthogonal Complement

The set of all vectors in

V

that are orthogonal to every vector in a subspace

W

W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v}, \mathbf{w} \rangle = 0 \text{ for all } \mathbf{w} \in W\}

Orthogonality

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Together,

W

and

W^\perp

span the entire space, with no overlap except the zero vector.

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Orthogonal Matrix

A square matrix

Q

satisfying

Q^TQ = QQ^T = I

Matrix Types

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A matrix whose columns (and rows) form an orthonormal set. Orthogonal matrices preserve lengths and angles — they represent rotations and reflections.

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