Linear map

Linear map
Tag	`0RJF668P1T`
URI	`m2://0RJF668P1T`
Kind	definition
Writing	W1 stated
MSC	15A04
Defines	Linear map
Uses	Vector space

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Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$ . A map $T \colon V \to W$ is linear when, for all $u, v \in V$ and $a \in F$ , $T(u + v) = T(u) + T(v), \qquad T(av) = a\, T(v).$ A linear map is also called a linear transformation.

Remarks

A linear map sends $0$ to $0$ and linear combinations to linear combinations: $T(\sum a_i v_i) = \sum a_i T(v_i)$ . For an $m \times n$ matrix $A$ , the map $F^n \to F^m$ , $x \mapsto Ax$ , is linear. Rotations of $\R^2$ 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.

Explanations

L1 human

A linear map respects the two operations of a vector space. The image of a sum is the sum of the images, and the image of a scaled vector is the scaled image. In the plane, linear maps send lines through the origin to lines through the origin, or to the origin, and keep parallel lines parallel.

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Vector space definition
Linear map definition

What depends on this

140 nodes use this node, directly or through other nodes.

Singular value decomposition 000NPTMX6J
Dimension of the dual space 00889YD15K
Linear map of a matrix 009892PZ30
Scaling a row scales the determinant 009CY8JXFX
Nullity 00K4FK4NBH
Jordan–Chevalley decomposition 011K11VPJN
Leibniz formula for determinants 016PE4VZ8Q
Determinant of a scalar multiple 01EXSMK864
Adding a multiple of one row to another preserves the determinant 01ZTQBK64Q
Reflection 025158VEYZ
Kernel of a linear map 02YFME3RGW
Triangularisation 0363JMQ42Y
Characteristic polynomial 039HF58KQ5
Determinant of a linear map 03F5KP94K0
A matrix is invertible iff its determinant is a unit 03Q7RE2HXR
Rational canonical form 042JWAR3Q0
Determinant is multiplicative 04CPJ6AF00
Spectrum of a matrix 04EHB7YDQS
Riesz representation theorem in finite dimensions 04GYPXDXNH
Determinant as signed volume 04X3PR72VJ
Product of a matrix and its adjugate 0528HKWZ1Z
Annihilating polynomial 052M4SX57M
Eigenvalues are the roots of the characteristic polynomial 053QVHF6N7
A matrix is diagonalisable iff it has a basis of eigenvectors 056SW1E0AC
Isomorphism with the coordinate space 059THXBRBB
Matrix of a composition is the product of matrices 05Q1B7MH0M
Composition of linear maps 05YBKMRVAR
Laplace expansion along a column 06AMCJMMVM
Spectral theorem 06J4PXRG22
Dimension of the annihilator 06Q1EFVH0Q
Geometric multiplicity 06YSJVR356
Linear isometry 0717XXPWM6
Determinant of the inverse 0783S9D0ZH
Generalised eigenspace 07PNDWA7CE
Existence of an eigenvalue over an algebraically closed field 08CJB2NQ92
Cauchy–Binet formula 08WE2W7DYS
An injective endomorphism of a finite-dimensional space is surjective 0912EEEVCT
Minimal polynomial divides the characteristic polynomial 093X6MN6Y6
Cramer's rule 0977KZ6PQB
Permuting rows multiplies the determinant by the sign 09YB3WQG38
Determinant of a matrix exponential 0A1QQN333A
Multiplication by a matrix as a linear map 0AJ0KP1B31
Eigenspace 0AYEPBQMAW
Eigenvector 0B5H4208Z9
Rotation matrix 0B9QNFGTHM
Matrix determinant lemma 0BC3Q6K92H
A linear map is injective iff its kernel is trivial 0BJBYCGMWE
Polynomial of an endomorphism 0BNEXSKNRV
Least squares 0BYTQ4X1HC
Characteristic polynomial of a linear map 0BZ81AJNWF
Laplace expansion along a row 0C08BQ3M05
Determinant is the product of the eigenvalues 0C2F40MEBC
Dual map 0C2TQMW5A1
Determinant 0C3CFHDN6R
Eigenvalues of a self-adjoint operator are real 0C4NT4GT3X
Special orthogonal group 0CGYNDCWRR
Matrix of the adjoint is the conjugate transpose 0DBTE6HVBD
Spectral theorem for Hermitian matrices 0DGZBQWZWN
Inverse in terms of the adjugate 0ENVM493VK
Normal operator 0ENZN5G4CV
Invariant subspace 0EP59ESQ94
Gershgorin circle theorem 0ETGP0XVCJ
Cayley–Hamilton theorem 0F765KFC2J
Trace is a coefficient of the characteristic polynomial 0FRYPZ6D0X
Determinant of a matrix with a zero row 0FS6B46E5T
Sylvester's determinant theorem 0G24VQ1088
Adjugate matrix 0GMPYM0NSR
Rule of Sarrus 0H8SZFFDXJ
Eigenvectors with distinct eigenvalues are linearly independent 0HNMFG8F8K
Eigenvalues of a positive definite matrix are positive 0HYC2RHD3M
Determinant of a block triangular matrix 0J9TVTNB4K
Image of a linear map 0JB228DCWB
Rank of a linear map 0JGB7A6GTS
Eigenspaces of a self-adjoint operator are orthogonal 0JHGAV9H25
Determinant of a triangular matrix 0K1KKMFHYH
Singular value 0K2TX12X58
Determinant of a matrix with two equal rows 0K4VQB7HWR
Rayleigh quotient 0K4WCNTMHG
Determinant via the Schur complement 0KA7A4ENVJ
Determinant of a linear map is the determinant of its matrix 0KMN32XGPF
Dual space 0M856Q1AMZ
Nilpotent endomorphism 0M8MHYERTY
Self-adjoint operator 0MB8HQW495
Orthogonal projection is the closest point 0MCDDSCB9Z
Diagonalisable iff the minimal polynomial has distinct roots 0MPAJRB23T
Change of basis formula 0MQXVZEBMT
Alternating multilinear map 0MSWYWHYF5
First isomorphism theorem for linear maps 0MTBGVZ4XJ
Trace is the sum of the eigenvalues 0NMXDCNNKR
Vandermonde determinant 0P84B554AS
Restriction of an endomorphism to an invariant subspace 0PFPWKBDTJ
Determinant is multilinear in the rows 0PPG793642
Null space 0PQ2GGY1QV
Principal axis theorem 0PR5P9Z4KM
Endomorphism 0PTF1PTJK0
Orientation of a real vector space 0Q8BZXVT1Z
Orthogonal projection 0Q8HZTJC7G
Cayley–Hamilton theorem for linear maps 0QEM68Y0DE
Eigenvalues are the roots of the minimal polynomial 0QF4ZEFG52
Semisimple endomorphism 0QS8P74F2Z
Annihilator of a subspace 0R6M158SQ9
A linear isometry preserves the inner product 0R7VXTYHYZ
A linear map is determined by its values on a basis 0R8X9XA83A
Projection 0RH4E3CAV9
Trace of a linear map 0RPDTWY2JS
Linear form 0RT96J4D3H
Algebraic multiplicity 0S7A0EMPF4
Generalised eigenvector 0SAAHC36J9
Minor 0SD528P0EK
Multilinear map 0SDMPMCN43
Differentiation as a linear map 0SS770NS8B
Adjoint of a linear map 0ST8QF9T3T
Determinant is a coefficient of the characteristic polynomial 0T0X8GTYYS
Characteristic polynomial does not depend on the basis 0TYMASG33Y
Minimal polynomial 0V58FPM3EQ
Determinant of a family of vectors in a basis 0V94H674MB
Special linear group 0V9PMN0RBH
Companion matrix 0VH1TC1PWF
Finite-dimensional spaces are isomorphic iff they have equal dimension 0VJBSK15SN
Cartan–Dieudonné theorem 0VMWGYY35Q
Space of linear maps 0VPCJWGMP9
Minimal polynomial divides every annihilating polynomial 0VTAD25A3G
Cofactor 0W2QAVJQ1W
Rank–nullity theorem 0W4FSW1TVB
Dual basis 0W6M3PSD08
Trace of a linear map is the trace of its matrix 0WCPH5Z7FA
Sylvester's criterion 0WKEF78PMV
Linear isomorphism 0XSJ25Z9M4
Distinct eigenvalues imply diagonalisable 0Y95Z7MT02
Universal property of the tensor product 0YDRV01R8F
Matrix of a linear map 0YYMY95FWS
Decomposition into generalised eigenspaces 0Z3GDY89T5
Determinant of the identity matrix 0Z4BWARC38
Dimension of the space of linear maps 0Z9N1ER0W0
Universal property of the quotient space 0ZERJX6PH6
Canonical isomorphism with the double dual 0ZJM9D7NPS
Determinant of a 2 by 2 matrix 0ZN2J42SK7
Eigenvalue 0ZND7NQ970
Matrix of the dual map is the transpose 0ZPB3180AY
Determinant of a transpose 0ZZQ2E1MTE

Sources

Web https://en.wikipedia.org/wiki/Linear_map, Definition and first consequences

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Used by

Kernel of a linear map 02YFME3RGW
Composition of linear maps 05YBKMRVAR
Linear isometry 0717XXPWM6
Multiplication by a matrix as a linear map 0AJ0KP1B31
A linear map is injective iff its kernel is trivial 0BJBYCGMWE
Dual map 0C2TQMW5A1
Image of a linear map 0JB228DCWB
Endomorphism 0PTF1PTJK0
A linear map is determined by its values on a basis 0R8X9XA83A
Linear form 0RT96J4D3H
Multilinear map 0SDMPMCN43
Differentiation as a linear map 0SS770NS8B
Adjoint of a linear map 0ST8QF9T3T
Space of linear maps 0VPCJWGMP9
Linear isomorphism 0XSJ25Z9M4
Universal property of the tensor product 0YDRV01R8F
Matrix of a linear map 0YYMY95FWS
Universal property of the quotient space 0ZERJX6PH6