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 $\sequence {a_n}$ be a linearly independent sequence of vectors of $G$.

Let $\sequence {b_n}$ be a sequence of vectors of $H$.

Then there is a unique linear transformation $\phi: G \to H$ satisfying:

$\forall k \in \closedint 1 n: \map \phi {a_k} = b_k$

Proof

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

The result then follows from Unique Linear Transformation Between Modules.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Theorem $28.4$: Corollary