Calculating L(x) of vectors in the Span

• We have a linear tf: $L:R^n\to R^m$.
• We have a set $S=\{{\vec{x}}_1,...,{\vec{x}}_k\}\in R^n$
• For each ${\vec{x}}_i$, we know $L({\vec{x}}_i)$
• Since every vector in Span(S) is just a linear combination of the vectors in S and since $L$ preserves linear combinations, we can calculate $L$ for every vector in the span.
• The proof/example in the reference doesn’t make sense.