Linear maps uniqueness proof: Difference between "uniquely determined on span($v_1,...,v_n$)" and "uniquely determined on V"

I've just been introduced to linear maps, and I'm still trying to wrap my head around the proofs, which don't read easily for me. In the uniqueness proof, the very last section states:

Thus $T$ (the linear map) is uniquely determined on span($v_1,...,v_n$) by the equation above. Because $v_1,...,v_n$ is a basis of V, this implies that T is uniquely determined on $V$.

My question is, what is the difference between those two sentences? If $T$ is uniquely determined on the span of a vector space, isn't it already uniquely determined on that space? My guess is that my confusion stems from not having a good understanding of what "uniquely determined" really means, so clarification on that would help as well.

• Sure, but when one speaks of the span of a list of vectors, it does not mean this list is a basis. Jul 27 '16 at 23:24
• Wow, can't believe I missed that. Thank you. Jul 27 '16 at 23:40
• It's often a matter of phrasing! Jul 27 '16 at 23:43

As Bernard pointed out in the comments, the author's intent was to separate the notion that $T$ was uniquely determined on the span of any list of vectors $v_1,...,v_n$, and the specific list in question was a basis of $V$, and so $T$ was uniquely determined on $V$.