I ran into this question as I was trying to solve a different problem, but I thought it merited some thought and was wondering if anyone might have an answer. Quite frankly, I'm confusing myself and don't even know if what I'm thinking about makes sense.
A linear functional is a linear map from a vector space to a scalar. Then if we define a basis $B = $ {$v_1, v_2, ..., v_n$}, we can define our linear functional, call it $\phi$, as a linear combination of the dual basis. Will the value of this linear functional on a vector $v$ change if we transform $v$ to its representation in a different basis? Basically, what I'm picturing is $\phi(v) = c_1v_1^*(v) + \dots + c_nv_n^*(v)$, which is easy to handle if we write $v$ in terms of $B$. If we were to write $v$ in terms of another basis, say $B'$, would $\phi([v]_{B'})$ = $\phi([v]_B)$?
Another thought I had on the heels of that was then if we defined $\phi$ by the dual basis of whatever basis we are representing $v$ in, does that change anything? (for example if we have $[v]_{B'}$ instead of $[v]_B$, can we redefine $\phi$ as a linear combination of $B'^*$ organically and does it affect anything?) I'm beginning to think maybe there's a tensor hidden in my question somewhere but I'm not quite sure how to make use of it (now thinking about it, $V \otimes V^*$ might be what I'm getting at, though I'm really not sure).
I'm generally just really confused. Any and all input (including telling me my question makes no sense with some clarification as to why) is welcome.
Your question makes no sense. Note that $\phi$ is a linear map from a vector space $V$ into the field $k$ that you are working with. Then you ask whether or not $\phi([v]_{B'})=\phi([v]_B)$. But, unless $V=k^n$, $[v]_B$ and $[v]_{B'}$ are not elements of $V$; they are elements of $k^n$. So, it makes no sense to talk about $\phi([v]_{B'})$ and $\phi([v]_B)$.