A [linear map,](Linear%20map.md) $A,$ is also considered to be a _linear operator_ if $A$ maps elements of some vector space $\mathcal{V}$ to other elements in that vector space. That is, a linear map is a linear operator if that operator is defined as $A:\mathcal{V}\rightarrow \mathcal{V}.$ ^8e4d93 # Sets of linear operators We express the [set](Sets.md) of [linear operators](Linear%20operator.md) on [vector space](Vector%20spaces.md) $\mathcal{V}$ as a [set of linear maps](Linear%20map.md#Sets%20of%20linear%20maps) from $\mathcal{V}$ to $\mathcal{V}$ as $\mathcal{L}(\mathcal{V},\mathcal{V})=\{A:\mathcal{V}\rightarrow \mathcal{V}\},$ which we shorten to $\mathcal{L}(\mathcal{V})$ where $A\in\mathcal{L}(\mathcal{V})$ is some linear operator in the set. ^af4198 #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators