I just started learning about linear maps and I had a quick question about the following step from a proof outlining the existence of a unique linear map $T: V \rightarrow W$.
Note: $L(V, W)$ below denotes the set of all linear maps from $V$ to $W$.
Suppose $T \in L(V,W)$ and $Tv_j = w_j$ for $j = 1, \dots, n$.
Let $c_1, \dots, c_n \in F$. The homogeneity of $T$ implies that
$T(c_j v_j) = c_j w_j$ for $j = 1, \dots, n$. The additivity of $T$
now implies that $$T(c_1 v_1 + \dots + c_n v_n) = c_1 w_1 + \dots + c_n w_n.$$Thus $T$ is uniquely determined on span$(v_1, \dots, v_n)$ by the
equation above.
What does it mean above that "$T$ is uniquely determined on span"? I understand that homogeneity and additivity hold for linear maps (by definition) but I have no idea how we followed those properties in the proof with "unique determination of $T$".
Best Answer
It means that if $Q$ is another linear map from $V$ into $W$ and if the condition$$(\forall j\in\{1,2,\ldots,n\}):Qv_j=w_j$$also holds, then $Q=T$.