Linear maps

A linear map \(T\) is a function (= map, mapping) between two vector spaces \(V\) and \(W\) over a field \(\mathbb{F}\): \[\begin{equation} \require{physics} T: V \rightarrow W \thinspace , \end{equation}\] such that \(\forall \vb{v}_1, \vb{v}_2 \in V; \forall a \in \mathbb{F}:\) \[\begin{align} &T(\vb{v}_1 + \vb{v}_2) = T(\vb{v}_1) + T(\vb{v}_2) && \text{$T$ `preserves' vector addition} \\ &T(a \vb{v}_1) = a T(\vb{v}_1) && \text{$T$ `preserves' scalar multiplication} \end{align}\]

We will call the set of all linear maps from \(V\) to \(W\) \(\mathcal{L}(V, W)\). If we define the sum of two linear maps \(S\) and \(T\) and the scalar product of an element \(a \in \mathbb{F}\) with a linear map \(T\) such that \(\forall \vb{v} \in V:\) \[\begin{align} &(S + T)(\vb{v}) = S(\vb{v}) + T(\vb{v}) \\ &(aS)(\vb{v}) = a(S(\vb{v})) \thinspace , \end{align}\] respectively, we can show that \(\mathcal{L}(V, W)\) forms a vector space over the field \(\mathbb{F}\).

A linear operator is a linear map \(T\) from a vector space \(V\) to itself: \[\begin{equation} T: V \rightarrow V \thinspace . \end{equation}\] Obviously, \(\mathcal{L}(V) = \mathcal{L}(V, V)\) also forms a vector space over \(\mathbb{F}\).