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.

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

[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 ${\textstyle \Lambda :X\to Y}$ whose proofs are so easy that we omit them; it is assumed that ${\textstyle A\subset X}$ and ${\textstyle B\subset Y}$ :
${\textstyle \Lambda 0=0.}$
If $A$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda (A)}$
If $B$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda ^{-1}(B)}$
In particular, the set: $\Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )$ is a subspace of $X$ , called the null space of ${\textstyle \Lambda }$ .

[3] ${\textstyle \Lambda 0=0.}$

[4] If $A$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda (A)}$

[5] If $B$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda ^{-1}(B)}$

[6] In particular, the set: $\Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )$ is a subspace of $X$ , called the null space of ${\textstyle \Lambda }$ .

[7] ${\textstyle \Lambda 0=0.}$

[8] If $A$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda (A)}$

[9] If $B$ is a subspace (or a convex set, or a balanced set) the same is true of ${\textstyle \Lambda ^{-1}(B)}$

[10] In particular, the set: $\Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )$ is a subspace of $X$ , called the null space of ${\textstyle \Lambda }$ .

[1]

[2]