I'm stuck on a few ideas from Hoffman and Kunze's Linear Algebra text Sec 3.5.
Let $V$ be a finite vector space over a field $F$ and let $f$ be a linear transformation $f: V \to F$.
There are a few properties I'm trying to work through. In particular, the ideas at the bottom of page 98 continuing on to page 9:
Where I'm stuck is the following. If I have a vector space $V$. Let's say $V$ has basis $B=\{\beta_1,...,\beta_k\}.$ Then for a vector $v$ we have $v=\sum_{i=1}^{k}c_i\beta_i$.
Would it be accurate to say that $f(v)=f(\sum_{i=1}^{k}c_i\beta_i)=\sum_{i=1}^{k}c_if_i(\beta_i)=\sum_{i=1}^{k}c_i$
What you denota as $c$ is actually denoted as $\alpha$ in the text, so I will use that notation:
Let $\{\alpha_i\}_i$ a basis of the vector space $V$, the dual basis will be denoted $\{f_i\}_i$. Then for the vector $v=\sum_i a_i \alpha_i$ we get:
$$f(v)=f(\sum_{i=1}^{k}a_i\alpha_i)=\sum_{i=1}^{k}a_if(\alpha_i)$$
However, now we can expand $f$ in the dual basis: $f=\sum_j c_j f_j$.
Then $$f(v) = \sum_{i=1}^{k}a_if(\alpha_i) = \sum_{i,j=1}^{k}a_i c_jf_j(\alpha_i)= \sum_{i,j=1}^{k}c_j a_i\delta_{ij} = \sum_i c_ia_i. $$