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linear transformation (Definition)

Let $ V$ and $ W$ be vector spaces over the same field $ F$. A linear transformation is a function $ T\colon V \to W$ such that:

  • $ T(v+w) = T(v)+T(w)$ for all $ v,w \in V$
  • $ T(\lambda v) = \lambda T(v)$ for all $ v\in V$, and $ \lambda \in F$
The set of all linear maps $ V \to W$ is denoted by $ \operatorname{Hom}_F(V,W)$ or $ \mathscr{L}(V,W)$.

Examples:

Properties:

  • $ T(0) = 0$.
  • If $ S$ and $ T$ are linear transformations from $ V$ to $ W$, and $ k\in F$, then so are $ S+T$ and $ kT$. As a result, $ \operatorname{Hom}_F(V,W)$ is a vector space over F.
  • If $ G\colon W\to U$ is a linear transformations then $ G\circ T\colon V\to U$ is also a linear transformation.
  • The kernel $ \operatorname{Ker}(T)=\{v\in V \mid T(v) = 0\}$ is a subspace of $ V$.
  • The image $ \operatorname{Im}(T) = \{T(v) \mid v\in V\}$ is a subspace of $ W$.
  • The inverse image $ T^{-1}(w)$ is a subspace if and only if $ w=0$.
  • A linear transformation is injective if and only if $ \operatorname{Ker}(T)=\{0\}$.
  • If $ v \in V$ then $ T^{-1}(T(v)) = v + \operatorname{Ker}(T)$.
  • If $ w\in \operatorname{Im}(T)$ then $ T(T^{-1}(w)) = \{w\}$.

Remark. A linear transformation $ T:V\to W$ such that $ W=V$ is called a linear operator, and a linear functional when $ W=F$.

See also:



"linear transformation" is owned by CWoo. [ full author list (4) | owner history (3) ]
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See Also: matrix, invariant subspace, dual homomorphism, kernel of a linear mapping, eigenvalue, nilpotent transformation, affine transformation, subadditive, matrix representation of a linear transformation

Other names:  linear map, vector space homomorphism, linear mapping
Also defines:  linear operator

Attachments:
linear transformation is continuous if its domain is finite dimensional (Theorem) by matte
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Cross-references: Wikipedia, linear functional, injective, inverse image, subspace, derivative, continuous functions, differentiable functions, vector, multiplication, matrix, function, field, vector spaces
There are 225 references to this entry.

This is version 20 of linear transformation, born on 2001-11-07, modified 2008-01-13.
Object id is 697, canonical name is LinearTransformation.
Accessed 44011 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
Pending Errata and Addenda
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Discussion
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forum policy
Terminology by rmilson on 2002-02-14 22:00:00
While in principle, the terms "map", "mapping",
"function", "transformation", etc are synonyms,
my impression is that the different terms have acquired
distinct meanings. This is more a matter of connotation than
denotation. However consistent usage that conforms to
prevalent norms should make for clearer communication.

The generic term is "mapping", or "map" - although
mapping seems to be the preferred term.

The word "function" should be reserverd for a "mapping"
whose domain and codomain are sets of "numbers" in
some general sense.

The word "transformation" should be reseverd for a
"mapping" where the domain and codomain coincide.
The basic idea is that one can compose a transformation
with itself.

The word "operator" should be reserved for "transformations"
whose domain/codomain is a set of "functions".

The word "functional" should be reserved for a mapping whose
domain is a set of "functions" and whose codomain is a set of
"numbers", in some general sense.
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