A linear map preserves linear combinations and has a matrix once bases are chosen.
Key knowledge
A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
Values and equations
T(αu+βv)=αT(u)+βT(v); yᵢ=Σⱼ₌₁ⁿ aᵢⱼxⱼ
T
Linear map from ℝⁿ to ℝᵐ
α, β
Real scalars
u, v
Input vectors
aᵢⱼ
Entry of matrix A in row i and column j
xⱼ
Input coordinate j
yᵢ
Output coordinate i
n, m
Positive input and output dimensions
Conditions and scope
α and β are real scalars and u,v are input vectors.
The matrix depends on the chosen bases; x and y are coordinate column vectors in those bases.
Matrix-vector dimensions must match; n input coordinates produce m output coordinates.
Sources · 4
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Real finite-dimensional linearity, basis dependence and matrix-vector multiplication.
2026-10-10 · Checked the linear-combination rule and basis construction against MIT, and compatible dimensions and entry computation against OpenStax.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Linearity preserves sums and scaling; real matrix maps act on coordinate columns.
The inspected matrix examples use standard coordinates; arbitrary-basis representation is a broader repository statement.
A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
Every real m×n matrix A admits A=UΣVᵀ with orthogonal U,V and rectangular diagonal Σ having nonnegative entries. These singular values are square roots of eigenvalues of AᵀA, with zero padding as needed.
A real orthogonal square matrix Q satisfies QᵀQ=I and preserves Euclidean lengths. For full-column-rank A∈Rᵐˣⁿ, m≥n, a thin QR factorization A=QR has orthonormal Q columns and nonsingular upper-triangular R.
Swapping rows, scaling a row by a nonzero scalar and adding a multiple of another row preserve a linear system's solution set. Reduction to echelon form followed by back substitution identifies pivots, free variables and inconsistent equations 0=c≠0.
With Aᵢⱼ formed by deleting row i and column j, cofactor expansion gives det A=Σⱼ(−1)ⁱ⁺ʲaᵢⱼdet Aᵢⱼ, starting from det[a]=a. It satisfies det(AB)=det A det B; A is invertible exactly when det A≠0.
For a linear equation Ly=f, any one particular solution yₚ plus the general homogeneous solution y_c gives every solution y=y_c+yₚ. Undetermined coefficients tries polynomial/exponential/trigonometric forcing forms, multiplying by sufficient powers of t when the trial overlaps a homogeneous mode.
A chain satisfies P(Xₙ₊₁=j|X₀,…,Xₙ)=Pᵢⱼ when Xₙ=i, where P has nonnegative entries and each row sums to one. With row-vector initial law μ, the law after n steps is μPⁿ.
Over F₂ take H=[A|Iₘ] and G=[Iₙ₋ₘ;A]. Encoding message x as c=Gx is injective, HG=0, and the code is exactly ker H. The first n−m coordinates hold information and the final m supply parity checks.
If columns of invertible P are an ordered basis B in standard coordinates, v=P[v]B. A linear operator with standard matrix A has basis matrix [T]B=P⁻¹AP. A new matrix representation changes coordinates, not the underlying map.
An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible. It preserves affine combinations whose weights sum to one, collinearity, parallelism and the signed parameter along a line. General affine transformations need not preserve Euclidean lengths or angles.
For N complex coefficients aⱼ define yₖ=Σ_(j=0)^(N−1)aⱼωᴶᴷ, with ω=e²πⁱ⁄ᴺ. The inverse is aⱼ=N⁻¹Σ_(k=0)^(N−1)yₖω⁻ᴶᴷ. This converts polynomial coefficients to values at the Nth roots of unity.
For a finite square matrix A over a field, its characteristic polynomial p(t)=det(tI−A)=tⁿ+cₙ₋₁tⁿ⁻¹+⋯+c₀ satisfies p(A)=Aⁿ+cₙ₋₁Aⁿ⁻¹+⋯+c₀I=0. Polynomial division by p reduces any polynomial in A to degree below n; when c₀≠0 the identity also expresses A⁻¹ as a polynomial in A.
For a locally planar bed with true dip δ, choose horizontal x toward its down-dip direction and elevation z upward. Its plane satisfies z=z₀−x tanδ. A vertical section making horizontal angle β to strike has apparent dip magnitude α with tanα=tanδ|sinβ|. A section normal to strike displays the true dip; a section along strike displays no dip for a nonvertical bed.
Projection uses a coordinate representation of a plane.
Three-component compositions occupy a normalized triangle: x_A+x_B+x_C=1 with nonnegative fractions. In a three-phase equilibrium, the bulk composition is x₀=f₁x₁+f₂x₂+f₃x₃ with f₁+f₂+f₃=1. Nonnegative phase fractions express the bulk point as barycentric coordinates of the tie triangle. Two-phase mixtures lie on their tie line.
Independent component equations determine the phase fractions.
A linear map T:X→Y between normed spaces is continuous exactly when ||Tx|| ≤ C||x|| for some finite C. Its operator norm is sup over unit vectors of ||Tx||. The continuous dual X* contains bounded linear functionals; arbitrary algebraic functionals need not be continuous in infinite dimensions. If Y is Banach then bounded operators X→Y form a Banach space in this norm.
Finite-dimensional linear maps are the bounded special case.
A covector is a linear functional on the tangent space, and a tensor is multilinear on tangent/cotangent inputs. For a smooth map F:M→N the tangent map dF pushes vectors forward at a point, while covectors and differential forms pull back by composition with dF. The coordinate transformation rule is essential; a table of components alone is not an invariant tensor.
A representation of a finite group G is a homomorphism G→GL(V). If the field characteristic does not divide |G|, every invariant subspace in a finite-dimensional representation has an invariant complement: average an arbitrary projection over the group. In characteristic p dividing |G| this averaging denominator is unavailable and complete reducibility can fail.