Linear Functional - Basis change Having a trouble understanding what you have to do in case you have a "simple" linear functional. The basis change and stuff for Quadratic and Bilinear forms are everywhere in every book, but there is not even a starting part for the linear functionals and their basis change. The question I have is of the style :
Given the functional $f[x,  y, z]^T = 3x - 2y + z$ in $\mathbb R^3$. Write the functional with respect to the coordinates of the basis : $(v_1,v_2,v_3)$ where $v_1 = [1,1,0]^T$, $v_2 = [0,2,1]^T$, $v_3 = [-2,1,1]^T$.
Now, the thought I had about this is to express the functional values of the ordinary basis $f(e_1),f(e_2),f(e_3)$ with respect to constants $a_{ij}$ and $e_i$. After that, I'd go on by expressing the vectors of the new basis with respect to $b_{ij}$ and $e_i$, to find the $b_{ij}$ and thus the matrix of the functional with respect to the "new" basis. Is that method correct ? I would really appreciate a hint or someone going through a complete solution of the given problem. Thanks in advance.

To find the matrix representation with respect to the new basis, we note that
$$
f(a_1 v_1 + a_2 v_2 + a_3 v_3) =\\
a_1f(v_1) + a_2f(v_2) + a_3f(v_3) = \\
a_1 - 3a_2 - 7a_3
$$
I would say that this equation constitutes an answer to the question as stated.
We can say moreover that the matrix of the linear functional with respect to this new basis is $\pmatrix{1&-3&-7}$.
