Space of linear maps

Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$. The space of linear maps $\Hom_F(V, W)$, also written $L(V, W)$, is the set of all linear maps $V \to W$. It has the operations $$(S + T)(v) = S(v) + T(v), \qquad (aT)(v) = a\, T(v)$$ for $S, T \in \Hom_F(V, W)$, $a \in F$ and $v \in V$. It is a vector space over $F$, with the zero map as zero vector.

Remarks

For $V = F^n$ and $W = F^m$, the matrix map is an isomorphism $\Hom_F(F^n, F^m) \cong M_{m \times n}(F)$. The dimension of the space of linear maps is $\dim V \cdot \dim W$ in finite dimensions. For $W = F$, it is the dual space of $V$, and for $W = V$, it is the space of endomorphisms of $V$.

Explanations

L1 human

Linear maps can be added and scaled, value by value, and the result is again linear. So the linear maps from $V$ to $W$ form a vector space themselves. In coordinates, this is the vector space of matrices of a fixed size.