Linear and multilinear algebra; matrix theory Definition
Linear map
Builds on Vector space
Definition
Let be a field and let and be vector spaces over . A map is linear when, for all and , A linear map is also called a linear transformation.
Remarks
A linear map sends to and linear combinations to linear combinations: . For an matrix , the map , , is linear. Rotations of about the origin are linear; translations by a nonzero vector are not linear. Each linear map between finite-dimensional spaces has a matrix in a choice of bases.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
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- Vector space definition
- Linear map definition
Where this leads
309 pages build on this one.
- Singular value decomposition
- Dimension of the dual space
- Linear map of a matrix
- Scaling a row scales the determinant
- Nullity
- Jordan–Chevalley decomposition
- Leibniz formula for determinants
- Determinant of a scalar multiple
- Adding a multiple of one row to another preserves the determinant
- Reflection
- Kernel of a linear map
- Triangularisation
- Characteristic polynomial
- Determinant of a linear map
- A matrix is invertible iff its determinant is a unit
- Rational canonical form
- Determinant is multiplicative
- Spectrum of a matrix
- Riesz representation theorem in finite dimensions
- Determinant as signed volume
- Product of a matrix and its adjugate
- Annihilating polynomial
- Eigenvalues are the roots of the characteristic polynomial
- A matrix is diagonalisable iff it has a basis of eigenvectors
- Isomorphism with the coordinate space
- Matrix of a composition is the product of matrices
- Composition of linear maps
- Laplace expansion along a column
- Spectral theorem
- Dimension of the annihilator
- Geometric multiplicity
- Linear isometry
- Determinant of the inverse
- Generalised eigenspace
- Existence of an eigenvalue over an algebraically closed field
- Cauchy–Binet formula
- An injective endomorphism of a finite-dimensional space is surjective
- Minimal polynomial divides the characteristic polynomial
- Cramer's rule
- Permuting rows multiplies the determinant by the sign
- Determinant of a matrix exponential
- Multiplication by a matrix as a linear map
- Eigenspace
- Eigenvector
- Rotation matrix
- Matrix determinant lemma
- A linear map is injective iff its kernel is trivial
- Polynomial of an endomorphism
- Least squares
- Characteristic polynomial of a linear map
- Laplace expansion along a row
- Determinant is the product of the eigenvalues
- Dual map
- Determinant
- Eigenvalues of a self-adjoint operator are real
- Special orthogonal group
- Matrix of the adjoint is the conjugate transpose
- Spectral theorem for Hermitian matrices
- Inverse in terms of the adjugate
- Normal operator
- Invariant subspace
- Gershgorin circle theorem
- Cayley–Hamilton theorem
- Trace is a coefficient of the characteristic polynomial
- Determinant of a matrix with a zero row
- Sylvester's determinant theorem
- Adjugate matrix
- Rule of Sarrus
- Eigenvectors with distinct eigenvalues are linearly independent
- Eigenvalues of a positive definite matrix are positive
- Determinant of a block triangular matrix
- Image of a linear map
- Rank of a linear map
- Eigenspaces of a self-adjoint operator are orthogonal
- Determinant of a triangular matrix
- Singular value
- Determinant of a matrix with two equal rows
- Rayleigh quotient
- Determinant via the Schur complement
- Determinant of a linear map is the determinant of its matrix
- Dual space
- Nilpotent endomorphism
- Self-adjoint operator
- Orthogonal projection is the closest point
- Diagonalisable iff the minimal polynomial has distinct roots
- Change of basis formula
- Alternating multilinear map
- First isomorphism theorem for linear maps
- Trace is the sum of the eigenvalues
- Vandermonde determinant
- Restriction of an endomorphism to an invariant subspace
- Determinant is multilinear in the rows
- Null space
- Principal axis theorem
- Endomorphism
- Orientation of a real vector space
- Orthogonal projection
- Cayley–Hamilton theorem for linear maps
- Eigenvalues are the roots of the minimal polynomial
- Semisimple endomorphism
- Annihilator of a subspace
- A linear isometry preserves the inner product
- A linear map is determined by its values on a basis
- Projection
- Trace of a linear map
- Linear form
- Algebraic multiplicity
- Generalised eigenvector
- Minor
- Multilinear map
- Differentiation as a linear map
- Adjoint of a linear map
- Determinant is a coefficient of the characteristic polynomial
- Characteristic polynomial does not depend on the basis
- Minimal polynomial
- Determinant of a family of vectors in a basis
- Special linear group
- Companion matrix
- Finite-dimensional spaces are isomorphic iff they have equal dimension
- Cartan–Dieudonné theorem
- Space of linear maps
- Minimal polynomial divides every annihilating polynomial
- Cofactor
- Rank–nullity theorem
- Dual basis
- Trace of a linear map is the trace of its matrix
- Sylvester's criterion
- Linear isomorphism
- Distinct eigenvalues imply diagonalisable
- Universal property of the tensor product
- Matrix of a linear map
- Decomposition into generalised eigenspaces
- Determinant of the identity matrix
- Dimension of the space of linear maps
- Universal property of the quotient space
- Canonical isomorphism with the double dual
- Determinant of a 2 by 2 matrix
- Eigenvalue
- Matrix of the dual map is the transpose
- Determinant of a transpose
- Non-negative least squares
- Pettis measurability theorem
- Conjugate residual method
- Rotation group SO(3)
- Horocycle flow
- List of simple Lie groups
- Local regression
- Beurling's theorem
- Wigner D-matrix
- Jacobi's formula
- Hilbert–Schmidt expansion theorem
- Minimal residual method
- Oscillator representation
- Eisenstein series
- Toeplitz operator
- Schwarz–Christoffel mapping
- Partial isometry
- Maass wave form
- Céa's lemma
- Modular lambda function
- The special linear group is the kernel of the determinant
- Modular curve
- Fuchsian group
- Pettis integral
- Representation theory of SL2(R)
- Compact symplectic group
- Siegel upper half-space
- Geodesic flow on a locally symmetric space
- Bilinear series of a symmetric kernel
- Ordinary least squares
- Lévy–Steinitz theorem
- Theta correspondence
- 6-j symbol
- Table of Lie groups
- Field trace
- Theta function
- Smoothing spline
- Eisenstein series as a double series
- Sylvester's determinant identity
- Ramanujan tau function
- Weil conjectures
- Lax–Milgram theorem
- Siegel modular form
- Gaussian curvature
- Metaplectic group
- Resultant
- Central polynomial
- Field norm
- Kaplansky's theorem on PI rings
- Principal curvature
- Shape operator
- Cayley–Menger determinant
- Poincaré group
- Compact operator
- Hausdorff paradox
- Kalman rank condition
- Response surface methodology
- Representation theory of SU(2)
- Nonlinear regression
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Generalized minimal residual method
- Minimum total potential energy principle
- 3-j symbol
- Selberg's 1/4 conjecture
- Arthur–Selberg trace formula
- Selberg trace formula
- Linear canonical transformation
- Arnold's cat map
- Natural transformation
- Arithmetic group
- Baily–Borel compactification
- Gauss–Newton algorithm
- Standard conjectures on algebraic cycles
- Schur polynomial
- Hilbert space adjoint
- Integral operator
- Trace form
- Margulis superrigidity theorem
- Wigner's classification
- SU(2)
- Congruence subgroup
- Lidskii's trace formula for integral operators
- Banach–Tarski paradox
- ADE singularity
- Wishart distribution
- Posner's theorem
- Inoue surface
- Spectrum of the Laplace–Beltrami operator
- Weyl–Titchmarsh theory
- Berlekamp's algorithm
- Topological quantum field theory
- SL(2,C)
- Hilbert projection theorem
- Spin group
- Eckart–Young theorem
- Ultraparallel theorem
- Zimmer's conjecture
- j-invariant
- Ridge regression
- Zermelo's navigation problem
- L-curve
- Dedekind eta function
- Image of J
- Modular form
- Fisher's inequality
- Biconjugate gradient stabilized method
- Kleinian group
- Exceptional Lie group E8
- Kovacic's algorithm
- Galerkin method
- Petersson inner product
- Jeffreys prior
- Operator monotone function
- Empirical risk minimization
- Grothendieck–Lefschetz trace formula
- Isotonic regression
- Projection body
- Mercer's theorem
- Nielsen–Thurston classification
- Horocycle
- Classical group
- Binary quadratic form
- SL(2,R)
- Lorentz group
- Wynn's epsilon algorithm
- Representation theory of the Lorentz group
- Derivation
- Principal component analysis
- Liouvillian function
- Poincaré half-plane model
- Differential Galois theory
- Shanks transformation
- Bidiagonalization
- Ladyzhenskaya–Babuška–Brezzi condition
- Cusp form
- Fundamental domain of the modular group
- Differential field
- Substitution tiling
- Topological modular forms
- Discriminant of a field extension
- Levenberg–Marquardt algorithm
- Schur product theorem
- Nome
- Riesz representation theorem
- Box–Jenkins method
- Singular values of a compact operator
- Pin group
- Lasso
- Householder transformation
- Numerical methods for linear least squares
- Masur–Veech theorem
- Modular group
- Schmidt's singular value decomposition of a kernel
- Rogers–Ramanujan continued fraction
- Pseudo-Anosov map
- Upper half-plane
- Hecke operator
- J-homomorphism
- Theta transformation formula
- Schramm–Loewner evolution
- Box spline
- de Branges space
- Amitsur–Levitzki theorem
- Hilbert's Theorem 90
- Nevanlinna parametrization of solutions
- Time-optimal control
- Loewner differential equation
- Exceptional Lie group G2
Further reading
- Web https://en.wikipedia.org/wiki/Linear_map, Definition and first consequences
About this page
| Tag | 0RJF668P1T |
|---|---|
| URI | m2://0RJF668P1T |
| Kind | definition |
| Writing | W1 stated |
| MSC | 15A04 |
| Defines | Linear map |
| Uses | Vector space |
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What this page needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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What links here
Used by
- Kernel of a linear map
- Composition of linear maps
- Linear isometry
- Multiplication by a matrix as a linear map
- A linear map is injective iff its kernel is trivial
- Dual map
- Image of a linear map
- Endomorphism
- A linear map is determined by its values on a basis
- Linear form
- Multilinear map
- Differentiation as a linear map
- Adjoint of a linear map
- Space of linear maps
- Linear isomorphism
- Universal property of the tensor product
- Matrix of a linear map
- Universal property of the quotient space
- Field trace
- Field norm
- Compact operator
- Integral operator
- Topological quantum field theory
- Derivation
- Substitution tiling