Unique Linear Transformation Between Vector Spaces
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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