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Linear Algebra Terms and Definitions

AAlgebraic MultiplicityEigenAAugmented MatrixLinear SystemsBBasisVector SpacesCChange of Basis MatrixTransformationsCCharacteristic PolynomialEigenCCofactorDeterminantsCCofactor Matrix (Adjugate)DeterminantsCColumn SpaceVector SpacesCCross ProductVectorsDDeterminantDeterminantsDDiagonal MatrixMatricesDDimensionVector SpacesDDot ProductVectorsEEigenspaceEigenEEigenvalueEigenEEigenvectorEigenGGeometric MultiplicityEigenHHomogeneous SystemLinear SystemsIIdentity MatrixMatricesIImage (Range)TransformationsIInner ProductOrthogonalityIInverse MatrixMatricesLLeft Null SpaceVector SpacesLLinear CombinationVectorsLLinear IndependenceVector SpacesLLinear TransformationTransformationsMMagnitude (Norm)VectorsMMatrixMatricesMMatrix RepresentationTransformationsMMinorDeterminantsNNull Space (Kernel)Vector SpacesOOrthogonal ComplementOrthogonalityOOrthogonal MatrixOrthogonalityOOrthogonal SetOrthogonalityOOrthogonal VectorsOrthogonalityOOrthonormal SetOrthogonalityPPivotLinear SystemsPPositive Definite MatrixMatricesRRankMatricesRReduced Row Echelon FormLinear SystemsRRow Echelon FormLinear SystemsRRow SpaceVector SpacesSScalarVectorsSSimilar MatricesTransformationsSSingular MatrixMatricesSSingular ValueEigenSSpanVector SpacesSSquare MatrixMatricesSSubspaceVector SpacesSSymmetric MatrixMatricesSSystem of Linear EquationsLinear SystemsTTraceMatricesUUnit VectorVectorsVVectorVectorsVVector 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
DeterminantMinorCofactorCofactor Matrix (Adjugate)
Eigen
EigenvalueEigenvectorEigenspaceCharacteristic PolynomialAlgebraic MultiplicityGeometric MultiplicitySingular Value
Linear Systems
System of Linear EquationsAugmented MatrixRow Echelon FormReduced Row Echelon FormPivotHomogeneous System
Matrices
MatrixSquare MatrixIdentity MatrixSymmetric MatrixInverse MatrixSingular MatrixRankTraceDiagonal MatrixPositive Definite Matrix
Orthogonality
Inner ProductOrthogonal VectorsOrthogonal SetOrthonormal SetOrthogonal ComplementOrthogonal Matrix
Transformations
Linear TransformationImage (Range)Matrix RepresentationChange of Basis MatrixSimilar Matrices
Vector Spaces
Vector SpaceSubspaceSpanLinear IndependenceBasisDimensionColumn SpaceNull Space (Kernel)Row SpaceLeft Null Space
Vectors
VectorScalarMagnitude (Norm)Unit VectorDot ProductCross ProductLinear Combination

55 terms

Vectors

(7 items)

Vector

An ordered list of nn real numbers: v=(v1,v2,…,vn)∈Rn\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n
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Scalar

An element of the underlying field — in standard linear algebra, a real number c∈Rc \in \mathbb{R}
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Magnitude (Norm)

∥v∥=v12+v22+⋯+vn2\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}
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Unit Vector

A vector u^\hat{\mathbf{u}} with ∥u^∥=1\|\hat{\mathbf{u}}\| = 1
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Dot Product

u⋅v=u1v1+u2v2+⋯+unvn=∥u∥ ∥v∥cos⁡θ\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = \|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta
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Cross Product

u×v=(u2v3−u3v2u3v1−u1v3u1v2−u2v1)\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}
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Linear Combination

c1v1+c2v2+⋯+ckvkc_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k

where c1,c2,…,ckc_1, c_2, \ldots, c_k are scalars
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Vector Spaces

(10 items)

Vector Space

A set VV equipped with vector addition and scalar multiplication satisfying the vector space axioms
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Subspace

A nonempty subset W⊆VW \subseteq V that is itself a vector space under the same operations
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Span

Span{v1,…,vk}={c1v1+⋯+ckvk∣ci∈R}\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}\}
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Linear Independence

Vectors v1,…,vk\mathbf{v}_1, \ldots, \mathbf{v}_k are linearly independent if
c1v1+⋯+ckvk=0  ⟹  c1=c2=⋯=ck=0c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \implies c_1 = c_2 = \cdots = c_k = 0
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Basis

A set {v1,…,vn}\{\mathbf{v}_1, \ldots, \mathbf{v}_n\} that is linearly independent and spans the entire vector space
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Dimension

dim⁡(V)=nwhere n is the number of vectors in any basis of V\dim(V) = n \quad \text{where } n \text{ is the number of vectors in any basis of } V
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Column Space

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

The span of the columns of AA
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Null Space (Kernel)

Nul(A)={x∈Rn∣Ax=0}\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}
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Row Space

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

The span of the rows of AA
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Left Null Space

Nul(AT)={y∈Rm∣ATy=0}\text{Nul}(A^T) = \{\mathbf{y} \in \mathbb{R}^m \mid A^T\mathbf{y} = \mathbf{0}\}
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Matrices

(10 items)

Matrix

A rectangular array of numbers with mm rows and nn columns: A∈Rm×nA \in \mathbb{R}^{m \times n}
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Square Matrix

A matrix with equal numbers of rows and columns: A∈Rn×nA \in \mathbb{R}^{n \times n}
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Identity Matrix

In=(10⋯001⋯0⋮⋮⋱⋮00⋯1)I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}
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Symmetric Matrix

A square matrix satisfying A=ATA = A^T
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Inverse Matrix

A square matrix A−1A^{-1} such that AA−1=A−1A=IAA^{-1} = A^{-1}A = I
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Singular Matrix

A square matrix AA with det⁡(A)=0\det(A) = 0
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Rank

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

tr(A)=a11+a22+⋯+ann=∑i=1naii\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}
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Diagonal Matrix

A square matrix where aij=0a_{ij} = 0 for all i≠ji \neq j
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Positive Definite Matrix

A symmetric matrix AA satisfying xTAx>0\mathbf{x}^T A \mathbf{x} > 0 for all nonzero x\mathbf{x}
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Determinants

(4 items)

Determinant

A scalar det⁡(A)∈R\det(A) \in \mathbb{R} assigned to every square matrix, defined recursively via cofactor expansion
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Minor

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

where A^ij\hat{A}_{ij} is the matrix obtained by deleting row ii and column jj
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Cofactor

Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}
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Cofactor Matrix (Adjugate)

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

where CC is the matrix of cofactors
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Linear Systems

(6 items)

System of Linear Equations

A 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}^m
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Augmented Matrix

[A∣b][A \mid \mathbf{b}]

The matrix AA with the vector b\mathbf{b} appended as an extra column
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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
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Reduced Row Echelon Form

Row echelon form with the additional requirements:
• every pivot is 11
• each pivot is the only nonzero entry in its column
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Pivot

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

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

(5 items)

Linear Transformation

A function T:V→WT: V \to W satisfying:
T(u+v)=T(u)+T(v)T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})

T(cu)=cT(u)T(c\mathbf{u}) = cT(\mathbf{u})
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Image (Range)

Im(T)={T(v)∣v∈V}\text{Im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}
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Matrix Representation

A matrix AA such that T(v)=A[v]BT(\mathbf{v}) = A[\mathbf{v}]_{\mathcal{B}} for a chosen basis B\mathcal{B}
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Change of Basis Matrix

A 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}}
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Similar Matrices

Matrices AA and BB are similar if B=P−1APB = P^{-1}AP for some invertible matrix PP
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Eigen

(7 items)

Eigenvalue

A scalar λ\lambda such that Av=λvA\mathbf{v} = \lambda\mathbf{v} for some nonzero vector v\mathbf{v}
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Eigenvector

A nonzero vector v\mathbf{v} such that Av=λvA\mathbf{v} = \lambda\mathbf{v} for some scalar λ\lambda
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Eigenspace

Eλ={v∈Rn∣Av=λv}=Nul(A−λI)E_\lambda = \{\mathbf{v} \in \mathbb{R}^n \mid A\mathbf{v} = \lambda\mathbf{v}\} = \text{Nul}(A - \lambda I)
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Characteristic Polynomial

p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I)
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Algebraic Multiplicity

The multiplicity of λ\lambda as a root of the characteristic polynomial
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Geometric Multiplicity

geo. mult.(λ)=dim⁡(Eλ)=dim⁡(Nul(A−λI))\text{geo. mult.}(\lambda) = \dim(E_\lambda) = \dim(\text{Nul}(A - \lambda I))
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Singular Value

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

where λi\lambda_i are the eigenvalues of ATAA^TA
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Orthogonality

(6 items)

Inner Product

A function ⟨⋅,⋅⟩:V×V→R\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R} satisfying symmetry, linearity, and positive-definiteness
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Orthogonal Vectors

Vectors u\mathbf{u} and v\mathbf{v} are orthogonal if ⟨u,v⟩=0\langle \mathbf{u}, \mathbf{v} \rangle = 0
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Orthogonal Set

A 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 j
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Orthonormal Set

An orthogonal set where every vector is a unit vector: ⟨vi,vj⟩=δij\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}
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Orthogonal Complement

W⊥={v∈V∣⟨v,w⟩=0 for all w∈W}W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v}, \mathbf{w} \rangle = 0 \text{ for all } \mathbf{w} \in W\}
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Orthogonal Matrix

A square matrix QQ satisfying QTQ=QQT=IQ^TQ = QQ^T = I
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