Linear Transformation ○꠹

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Linear Transformation ○꠹｜Definition｜1st｜20251119205401-00-⌔

Linear map - Wikipedia

Linear map

In mathematics, and more specifically in linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an $m\times n$ matrix, which takes vectors in $n$ -dimensions into vectors in $m$ -dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars.

A linear map is a homomorphism of vector spaces.¹ Thus, a linear map $T:V\to W$ satisfies ⁠ $T(ax+by)=aTx+bTy$ ⁠, where $a$ and $b$ are scalars, and $x$ and $y$ are vectors (elements of the vector space ⁠ $V$ ⁠). A linear mapping always maps the origin of $V$ to the origin of ⁠ $W$ ⁠, and linear subspaces of $V$ onto linear subspaces in $W$ (possibly of a lower dimension);² for example, it maps a plane through the origin in $V$ to either a plane through the origin in ⁠ $W$ ⁠, a line through the origin in ⁠ $W$ ⁠, or just the origin in ⁠ $W$ ⁠. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

Printed 2026-06-28.

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${\Lambda }^{-1}(\{0\})=\{\mathbf{x}\in X:\Lambda \mathbf{x}=0\}=N(\Lambda )$

is a subspace of X, called the null space of ⁠ $\Lambda$ ⁠

Footnotes

1. In the language of category theory, linear maps are the morphisms of vector spaces. Restricted to the category of finite-dimensional vector spaces, they form a category equivalent to the one of matrices. ↩

2. Rudin 1991, p. 14
   Here are some properties of linear mappings $\Lambda :X\to Y$ whose proofs are so easy that we omit them; it is assumed that $A\subset X$ and $B\subset Y$: $\Lambda 0=0.$ If A is a subspace (or a convex set, or a balanced set) the same is true of $\Lambda (A)$ If B is a subspace (or a convex set, or a balanced set) the same is true of ${\Lambda }^{-1}(B)$ In particular, the set: ↩

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