In Chapter R we will take up the matter of representations fully, where Theorem VRRB will be critical for Definition VR. We will now motivate and prove a critical theorem that tells us how to “represent” a vector. This theorem could wait, but working with it now will provide some extra insight into the nature of linearly independent spanning sets. First an example, then the theorem.
Suppose that $V$ is a vector space and $B=\set{\vectorlist{v}{m}}$ is a linearly independent set that spans $V$. Let $\vect{w}$ be any vector in $V$. Then there exist unique scalars $a_1,\,a_2,\,a_3,\,\ldots,\,a_m$ such that \begin{equation*} \vect{w}=\lincombo{a}{v}{m}. \end{equation*}
The converse of Theorem VRRB is true as well, but is not important enough to rise beyond an exercise (see Exercise LISS.T51).
This is a very typical use of the hypothesis that a set is linearly independent — obtain a relation of linear dependence and then conclude that the scalars must all be zero. The result of this theorem tells us that we can write any vector in a vector space as a linear combination of the vectors in a linearly independent spanning set, but only just. There is only enough raw material in the spanning set to write each vector one way as a linear combination. So in this sense, we could call a linearly independent spanning set a “minimal spanning set.” These sets are so important that we will give them a simpler name (“basis”) and explore their properties further in the next section.