Vector
An ordered list of $n$ real numbers: $\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n$
Scalar
An element of the underlying field — in standard linear algebra, a real number $c \in \mathbb{R}$
Magnitude (Norm)
The length of a [vector](!/linear-algebra/definitions#vector), measured as its distance from the origin:
$$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$
Unit Vector
A vector $\hat{\mathbf{u}}$ with $\|\hat{\mathbf{u}}\| = 1$
Dot Product
An operation that takes two [vectors](!/linear-algebra/definitions#vector) and returns a [scalar](!/linear-algebra/definitions#scalar), computed by summing the products of corresponding components:
$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = \|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta$$
Cross Product
A binary operation on two vectors in $\mathbb{R}^3$ that produces a vector perpendicular to both inputs:
$$\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}$$
Linear Combination
A sum of [vectors](!/linear-algebra/definitions#vector), each multiplied by a [scalar](!/linear-algebra/definitions#scalar) coefficient:
$$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k$$
Vector Space
A set $V$ equipped with vector addition and scalar multiplication satisfying the [vector space axioms](!/linear-algebra/vector-spaces/axioms)
Subspace
A nonempty subset $W \subseteq V$ that is itself a [vector space](!/linear-algebra/definitions#vector_space) under the same operations
Span
The set of all [linear combinations](!/linear-algebra/definitions#linear_combination) of a given collection of vectors:
$$\text{Span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k \mid c_i \in \mathbb{R}\}$$
Linear Independence
Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent if the only solution to
$$c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$$
is $c_1 = c_2 = \cdots = c_k = 0$
Basis
A set $\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}$ that is [linearly independent](!/linear-algebra/definitions#linear_independence) and [spans](!/linear-algebra/definitions#span) the entire [vector space](!/linear-algebra/definitions#vector_space)
Dimension
The number of vectors in any [basis](!/linear-algebra/definitions#basis) of a [vector space](!/linear-algebra/definitions#vector_space) $V$, denoted $\dim(V)$
Column Space
The set of all vectors expressible as $A\mathbf{x}$ — equivalently, the [span](!/linear-algebra/definitions#span) of the columns of $A$:
$$\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}$$
Null Space (Kernel)
The set of all solutions to the [homogeneous system](!/linear-algebra/definitions#homogeneous_system) $A\mathbf{x} = \mathbf{0}$:
$$\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}$$
Row Space
The [span](!/linear-algebra/definitions#span) of the rows of a [matrix](!/linear-algebra/definitions#matrix), equivalently the [column space](!/linear-algebra/definitions#column_space) of its transpose:
$$\text{Row}(A) = \text{Col}(A^T)$$
Left Null Space
The [null space](!/linear-algebra/definitions#null_space) of the transpose $A^T$ — the set of all vectors $\mathbf{y}$ satisfying $A^T\mathbf{y} = \mathbf{0}$:
$$\text{Nul}(A^T) = \{\mathbf{y} \in \mathbb{R}^m \mid A^T\mathbf{y} = \mathbf{0}\}$$
Matrix
A rectangular array of numbers with $m$ rows and $n$ columns: $A \in \mathbb{R}^{m \times n}$
Square Matrix
A [matrix](!/linear-algebra/definitions#matrix) with equal numbers of rows and columns: $A \in \mathbb{R}^{n \times n}$
Identity Matrix
The [square matrix](!/linear-algebra/definitions#square_matrix) with $1$s on the main diagonal and $0$s elsewhere, denoted $I_n$:
$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$
Symmetric Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) satisfying $A = A^T$
Inverse Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$
Singular Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $A$ with $\det(A) = 0$
Rank
The number of [linearly independent](!/linear-algebra/definitions#linear_independence) columns (equivalently, rows) in a [matrix](!/linear-algebra/definitions#matrix):
$$\text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A))$$
Trace
The sum of the main diagonal entries of a [square matrix](!/linear-algebra/definitions#square_matrix):
$$\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}$$
Diagonal Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) where $a_{ij} = 0$ for all $i \neq j$
Positive Definite Matrix
A [symmetric matrix](!/linear-algebra/definitions#symmetric_matrix) $A$ satisfying $\mathbf{x}^T A \mathbf{x} > 0$ for all nonzero $\mathbf{x}$
Determinant
A scalar $\det(A) \in \mathbb{R}$ assigned to every [square matrix](!/linear-algebra/definitions#square_matrix), defined recursively via [cofactor](!/linear-algebra/definitions#cofactor) expansion
Minor
The [determinant](!/linear-algebra/definitions#determinant) of the submatrix obtained by deleting row $i$ and column $j$ from a [matrix](!/linear-algebra/definitions#matrix):
$$M_{ij} = \det(\hat{A}_{ij})$$
Cofactor
A signed [minor](!/linear-algebra/definitions#minor), with sign determined by the position $(i,j)$:
$$C_{ij} = (-1)^{i+j} M_{ij}$$
Cofactor Matrix (Adjugate)
The transpose of the matrix of [cofactors](!/linear-algebra/definitions#cofactor) of $A$:
$$\text{adj}(A) = C^T$$
System of Linear Equations
A collection of equations $A\mathbf{x} = \mathbf{b}$ where $A$ is an $m \times n$ [matrix](!/linear-algebra/definitions#matrix) and $\mathbf{b} \in \mathbb{R}^m$
Augmented Matrix
The [matrix](!/linear-algebra/definitions#matrix) formed by appending the right-hand side vector $\mathbf{b}$ as an additional column to the coefficient matrix $A$, written $[A \mid \mathbf{b}]$
Row Echelon Form
A matrix where:
• all zero rows are at the bottom
• each leading entry ([pivot](!/linear-algebra/definitions#pivot)) is to the right of the pivot in the row above
• all entries below each pivot are zero
Reduced Row Echelon Form
[Row echelon form](!/linear-algebra/definitions#row_echelon_form) with the additional requirements:
• every [pivot](!/linear-algebra/definitions#pivot) is $1$
• each pivot is the only nonzero entry in its column
Pivot
The first nonzero entry in each row of a matrix in [row echelon form](!/linear-algebra/definitions#row_echelon_form)
Homogeneous System
A [system of linear equations](!/linear-algebra/definitions#system_of_linear_equations) in which every equation equals zero: $A\mathbf{x} = \mathbf{0}$
Linear Transformation
A function $T: V \to W$ between [vector spaces](!/linear-algebra/definitions#vector_space) that preserves addition and scalar multiplication:
$$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$$
$$T(c\mathbf{u}) = cT(\mathbf{u})$$
Image (Range)
The set of all output vectors of a [linear transformation](!/linear-algebra/definitions#linear_transformation):
$$\text{Im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}$$
Matrix Representation
A [matrix](!/linear-algebra/definitions#matrix) $A$ such that $T(\mathbf{v}) = A[\mathbf{v}]_{\mathcal{B}}$ for a chosen [basis](!/linear-algebra/definitions#basis) $\mathcal{B}$
Change of Basis Matrix
A matrix $P$ that converts coordinates from one [basis](!/linear-algebra/definitions#basis) to another: $[\mathbf{v}]_{\mathcal{B}'} = P^{-1}[\mathbf{v}]_{\mathcal{B}}$
Similar Matrices
Matrices $A$ and $B$ are similar if $B = P^{-1}AP$ for some invertible matrix $P$
Eigenvalue
A scalar $\lambda$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero [vector](!/linear-algebra/definitions#vector) $\mathbf{v}$
Eigenvector
A nonzero vector $\mathbf{v}$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$
Eigenspace
The set of all [eigenvectors](!/linear-algebra/definitions#eigenvector) for a given [eigenvalue](!/linear-algebra/definitions#eigenvalue) $\lambda$, together with the zero vector — equivalently, the [null space](!/linear-algebra/definitions#null_space) of $(A - \lambda I)$:
$$E_\lambda = \text{Nul}(A - \lambda I)$$
Characteristic Polynomial
The polynomial whose roots are the [eigenvalues](!/linear-algebra/definitions#eigenvalue) of $A$, obtained by computing:
$$p(\lambda) = \det(A - \lambda I)$$
Algebraic Multiplicity
The multiplicity of $\lambda$ as a root of the [characteristic polynomial](!/linear-algebra/definitions#characteristic_polynomial)
Geometric Multiplicity
The [dimension](!/linear-algebra/definitions#dimension) of the [eigenspace](!/linear-algebra/definitions#eigenspace) associated with an [eigenvalue](!/linear-algebra/definitions#eigenvalue) $\lambda$:
$$\text{geo. mult.}(\lambda) = \dim(E_\lambda) = \dim(\text{Nul}(A - \lambda I))$$
Singular Value
A nonnegative scalar measuring how much a matrix stretches space along each principal direction, derived from the [eigenvalues](!/linear-algebra/definitions#eigenvalue) of $A^TA$:
$$\sigma_i = \sqrt{\lambda_i(A^TA)}$$
Inner Product
A function $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}$ satisfying symmetry, linearity, and positive-definiteness
Orthogonal Vectors
Vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$
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 $i \neq j$
Orthonormal Set
An [orthogonal set](!/linear-algebra/definitions#orthogonal_set) where every vector is a [unit vector](!/linear-algebra/definitions#unit_vector): $\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}$
Orthogonal Complement
The set of all vectors in $V$ that are [orthogonal](!/linear-algebra/definitions#orthogonal_vectors) to every vector in a [subspace](!/linear-algebra/definitions#subspace) $W$:
$$W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v}, \mathbf{w} \rangle = 0 \text{ for all } \mathbf{w} \in W\}$$
Orthogonal Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $Q$ satisfying $Q^TQ = QQ^T = I$
Vector
An ordered list of $n$ real numbers: $\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n$
Scalar
An element of the underlying field — in standard linear algebra, a real number $c \in \mathbb{R}$
Magnitude (Norm)
The length of a [vector](!/linear-algebra/definitions#vector), measured as its distance from the origin:
$$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$
Unit Vector
A vector $\hat{\mathbf{u}}$ with $\|\hat{\mathbf{u}}\| = 1$
Dot Product
An operation that takes two [vectors](!/linear-algebra/definitions#vector) and returns a [scalar](!/linear-algebra/definitions#scalar), computed by summing the products of corresponding components:
$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = \|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta$$
Cross Product
A binary operation on two vectors in $\mathbb{R}^3$ that produces a vector perpendicular to both inputs:
$$\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}$$
Linear Combination
A sum of [vectors](!/linear-algebra/definitions#vector), each multiplied by a [scalar](!/linear-algebra/definitions#scalar) coefficient:
$$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k$$
Vector Space
A set $V$ equipped with vector addition and scalar multiplication satisfying the [vector space axioms](!/linear-algebra/vector-spaces/axioms)
Subspace
A nonempty subset $W \subseteq V$ that is itself a [vector space](!/linear-algebra/definitions#vector_space) under the same operations
Span
The set of all [linear combinations](!/linear-algebra/definitions#linear_combination) of a given collection of vectors:
$$\text{Span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k \mid c_i \in \mathbb{R}\}$$
Linear Independence
Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent if the only solution to
$$c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$$
is $c_1 = c_2 = \cdots = c_k = 0$
Basis
A set $\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}$ that is [linearly independent](!/linear-algebra/definitions#linear_independence) and [spans](!/linear-algebra/definitions#span) the entire [vector space](!/linear-algebra/definitions#vector_space)
Dimension
The number of vectors in any [basis](!/linear-algebra/definitions#basis) of a [vector space](!/linear-algebra/definitions#vector_space) $V$, denoted $\dim(V)$
Column Space
The set of all vectors expressible as $A\mathbf{x}$ — equivalently, the [span](!/linear-algebra/definitions#span) of the columns of $A$:
$$\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}$$
Null Space (Kernel)
The set of all solutions to the [homogeneous system](!/linear-algebra/definitions#homogeneous_system) $A\mathbf{x} = \mathbf{0}$:
$$\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}$$
Row Space
The [span](!/linear-algebra/definitions#span) of the rows of a [matrix](!/linear-algebra/definitions#matrix), equivalently the [column space](!/linear-algebra/definitions#column_space) of its transpose:
$$\text{Row}(A) = \text{Col}(A^T)$$
Left Null Space
The [null space](!/linear-algebra/definitions#null_space) of the transpose $A^T$ — the set of all vectors $\mathbf{y}$ satisfying $A^T\mathbf{y} = \mathbf{0}$:
$$\text{Nul}(A^T) = \{\mathbf{y} \in \mathbb{R}^m \mid A^T\mathbf{y} = \mathbf{0}\}$$
Matrix
A rectangular array of numbers with $m$ rows and $n$ columns: $A \in \mathbb{R}^{m \times n}$
Square Matrix
A [matrix](!/linear-algebra/definitions#matrix) with equal numbers of rows and columns: $A \in \mathbb{R}^{n \times n}$
Identity Matrix
The [square matrix](!/linear-algebra/definitions#square_matrix) with $1$s on the main diagonal and $0$s elsewhere, denoted $I_n$:
$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$
Symmetric Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) satisfying $A = A^T$
Inverse Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$
Singular Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $A$ with $\det(A) = 0$
Rank
The number of [linearly independent](!/linear-algebra/definitions#linear_independence) columns (equivalently, rows) in a [matrix](!/linear-algebra/definitions#matrix):
$$\text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A))$$
Trace
The sum of the main diagonal entries of a [square matrix](!/linear-algebra/definitions#square_matrix):
$$\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}$$
Diagonal Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) where $a_{ij} = 0$ for all $i \neq j$
Positive Definite Matrix
A [symmetric matrix](!/linear-algebra/definitions#symmetric_matrix) $A$ satisfying $\mathbf{x}^T A \mathbf{x} > 0$ for all nonzero $\mathbf{x}$
Determinant
A scalar $\det(A) \in \mathbb{R}$ assigned to every [square matrix](!/linear-algebra/definitions#square_matrix), defined recursively via [cofactor](!/linear-algebra/definitions#cofactor) expansion
Minor
The [determinant](!/linear-algebra/definitions#determinant) of the submatrix obtained by deleting row $i$ and column $j$ from a [matrix](!/linear-algebra/definitions#matrix):
$$M_{ij} = \det(\hat{A}_{ij})$$
Cofactor
A signed [minor](!/linear-algebra/definitions#minor), with sign determined by the position $(i,j)$:
$$C_{ij} = (-1)^{i+j} M_{ij}$$
Cofactor Matrix (Adjugate)
The transpose of the matrix of [cofactors](!/linear-algebra/definitions#cofactor) of $A$:
$$\text{adj}(A) = C^T$$
System of Linear Equations
A collection of equations $A\mathbf{x} = \mathbf{b}$ where $A$ is an $m \times n$ [matrix](!/linear-algebra/definitions#matrix) and $\mathbf{b} \in \mathbb{R}^m$
Augmented Matrix
The [matrix](!/linear-algebra/definitions#matrix) formed by appending the right-hand side vector $\mathbf{b}$ as an additional column to the coefficient matrix $A$, written $[A \mid \mathbf{b}]$
Row Echelon Form
A matrix where:
• all zero rows are at the bottom
• each leading entry ([pivot](!/linear-algebra/definitions#pivot)) is to the right of the pivot in the row above
• all entries below each pivot are zero
Reduced Row Echelon Form
[Row echelon form](!/linear-algebra/definitions#row_echelon_form) with the additional requirements:
• every [pivot](!/linear-algebra/definitions#pivot) is $1$
• each pivot is the only nonzero entry in its column
Pivot
The first nonzero entry in each row of a matrix in [row echelon form](!/linear-algebra/definitions#row_echelon_form)
Homogeneous System
A [system of linear equations](!/linear-algebra/definitions#system_of_linear_equations) in which every equation equals zero: $A\mathbf{x} = \mathbf{0}$
Linear Transformation
A function $T: V \to W$ between [vector spaces](!/linear-algebra/definitions#vector_space) that preserves addition and scalar multiplication:
$$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$$
$$T(c\mathbf{u}) = cT(\mathbf{u})$$
Image (Range)
The set of all output vectors of a [linear transformation](!/linear-algebra/definitions#linear_transformation):
$$\text{Im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}$$
Matrix Representation
A [matrix](!/linear-algebra/definitions#matrix) $A$ such that $T(\mathbf{v}) = A[\mathbf{v}]_{\mathcal{B}}$ for a chosen [basis](!/linear-algebra/definitions#basis) $\mathcal{B}$
Change of Basis Matrix
A matrix $P$ that converts coordinates from one [basis](!/linear-algebra/definitions#basis) to another: $[\mathbf{v}]_{\mathcal{B}'} = P^{-1}[\mathbf{v}]_{\mathcal{B}}$
Similar Matrices
Matrices $A$ and $B$ are similar if $B = P^{-1}AP$ for some invertible matrix $P$
Eigenvalue
A scalar $\lambda$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero [vector](!/linear-algebra/definitions#vector) $\mathbf{v}$
Eigenvector
A nonzero vector $\mathbf{v}$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$
Eigenspace
The set of all [eigenvectors](!/linear-algebra/definitions#eigenvector) for a given [eigenvalue](!/linear-algebra/definitions#eigenvalue) $\lambda$, together with the zero vector — equivalently, the [null space](!/linear-algebra/definitions#null_space) of $(A - \lambda I)$:
$$E_\lambda = \text{Nul}(A - \lambda I)$$
Characteristic Polynomial
The polynomial whose roots are the [eigenvalues](!/linear-algebra/definitions#eigenvalue) of $A$, obtained by computing:
$$p(\lambda) = \det(A - \lambda I)$$
Algebraic Multiplicity
The multiplicity of $\lambda$ as a root of the [characteristic polynomial](!/linear-algebra/definitions#characteristic_polynomial)
Geometric Multiplicity
The [dimension](!/linear-algebra/definitions#dimension) of the [eigenspace](!/linear-algebra/definitions#eigenspace) associated with an [eigenvalue](!/linear-algebra/definitions#eigenvalue) $\lambda$:
$$\text{geo. mult.}(\lambda) = \dim(E_\lambda) = \dim(\text{Nul}(A - \lambda I))$$
Singular Value
A nonnegative scalar measuring how much a matrix stretches space along each principal direction, derived from the [eigenvalues](!/linear-algebra/definitions#eigenvalue) of $A^TA$:
$$\sigma_i = \sqrt{\lambda_i(A^TA)}$$
Inner Product
A function $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}$ satisfying symmetry, linearity, and positive-definiteness
Orthogonal Vectors
Vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$
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 $i \neq j$
Orthonormal Set
An [orthogonal set](!/linear-algebra/definitions#orthogonal_set) where every vector is a [unit vector](!/linear-algebra/definitions#unit_vector): $\langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij}$
Orthogonal Complement
The set of all vectors in $V$ that are [orthogonal](!/linear-algebra/definitions#orthogonal_vectors) to every vector in a [subspace](!/linear-algebra/definitions#subspace) $W$:
$$W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v}, \mathbf{w} \rangle = 0 \text{ for all } \mathbf{w} \in W\}$$
Orthogonal Matrix
A [square matrix](!/linear-algebra/definitions#square_matrix) $Q$ satisfying $Q^TQ = QQ^T = I$
