Looking for clarification for a proof that relates injectivity and surjectivity with linear(in)dependence and generating sets There's a proof in my book which says the following:

Assume $V$ and $W$, two $K$-vectorspaces. $B$ is a basis for $V$ and $\{w_b|b\in B\}\subseteq W$ a subset of $W$. Consider the linear mapping:
  $g:V\rightarrow W:\sum_{b\in B}\lambda_bb\rightarrow\sum_{b\in B}\lambda_bw_b$
  Then the following is true:
  (i) g is injective iff $\{w_b|b\in B\}$ is a linearly independent set in W.
  ... (similar statements for surjectivity and bijectivity)

My question is, what is meant by "$\{w_b|b\in B\}$"? What is meant by the fact that the index is an element of the basis for $V$?

This means a fixed association $b \to w_b \in W$ for all $b \in B$.
