Unique Linear Transformation Between Vector Spaces

Corollary to Unique Linear Transformation Between Modules

Let $G$ be a finite-dimensional $K$-vector space.

Let $H$ be a $K$-vector space (not necessarily finite-dimensional).

Let $\left \langle {a_n} \right \rangle$ be a linearly independent sequence of vectors of $G$.

Let $\left \langle {b_n} \right \rangle$ be a sequence of vectors of $H$.

Then there is a unique linear transformation $\phi: G \to H$ satisfying $\forall k \in \left[{1 \,.\,.\, n}\right]: \phi \left({a_k}\right) = b_k$

Proof

From Generator of Vector Space Contains Basis, $\left\{{a_1, \ldots, a_m}\right\}$ is contained in a basis for $G$.

The result then follows from Unique Linear Transformation Between Modules.

$\blacksquare$