### Linear maps: Linear maps

### The notion of linear map

In order to be able to compare vector spaces, we use mappings between vector spaces that respect the vector space structure, the so-called linear maps. We begin with the definition of a map.

MapLet #X# and #Y# be two (possibly the same) sets. A **map** or **mapping** #G:X\rightarrow Y# assigns to each element #x# of #X# exactly one element #G(x)#, sometimes also noted as #Gx#, of #Y#. An explicit expression for #G(x)# is also called the **mapping rule**. The element #G(x)# of #Y# is referred to as the **image** of #x# under #G#. The set #X# is called the **domain** and #Y# the **codomain** or **range** of #G#. For an element #y# of #Y#, the **(full) preimage** of #y# under #G# is the set of all elements #x# of #X# satisfying #G(x) = y#.

A map #G# with domain #X# and codomain #Y# is also indicated by #x\mapsto G(x)#, where #G(x)# is replaced by a mapping rule.

For each vector space #V# the map #I: V \rightarrow V# given by #I(\vec{v}) =\vec{v}# is a map, called the **identity** **map,** or **identity** on #V#. If we want to make the vector space #V# explicit, then we write it #I_V# rather than #I#.

If both #V# and #W# are real vector spaces, then the map #O: V \rightarrow W# given by #O(\vec{v})=\vec{0}# is a map, the so-called **zero map**. We also denote this map by #0# and sometimes by #0_V#.

Linear map

Let #V# and #W# be two (possibly the same) vector spaces. A map #L:V\rightarrow W# is called **linear** if, for all vectors #\vec{x},\vec{y}\in V# and all numbers #\alpha#, we have

\[

\begin{array}{rclr}L (\vec{x}+\vec{y})&=&L\vec{x}+L\vec{y}&\phantom{xx}\color{blue}{\text{sum rule}}\\

L (\alpha \vec{x})&=&\alpha( L\vec{x})&\phantom{xx}\color{blue}{\text{scalar rule}}

\end{array}

\]

By repeated application of the definition we see that the image of a linear combination is the same linear combination of the image vectors:

Linearity of a map

A map #L:V\rightarrow W# is linear if and only if, for all natural numbers #n#, all vectors #\vec{x}_1,\ldots,\vec{x}_n# in #V# and all numbers #\alpha_1,\ldots ,\alpha_n#,

\[

L \left(\,\sum_{i=1}^n\alpha_i\vec{x}_i\,\right)=\sum_{i=1}^n\alpha_iL\vec{x}_i

\]

Linear maps occur very frequently in practice, even though they are not always immediately recognized as such. The following examples illustrate this.

*sum rule*and scalar rule are basic properties of the derivative, which we indicate by #D(f)# for a differentiable function #f#:

\[

\begin{array}{rcl}

D(f+g) & =&D(f)+D(g) \\

D(\alpha\cdot f) & =&\alpha\cdot D(f)

\end{array}

\]

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