Finding a dual basis

This is one of my homework questions - I'm pretty sure I understand part of it.

Let $V=\Bbb R^3$, and define $f_1, f_2, f_3 \in V^*$ as follows: $$f_1(x,y,z) = x - 2y;\quad f_2(x,y,z) = x + y + z; \quad f_3(x,y,z) = y - 3z.$$ Prove that $\{f_1, f_2, f_3\}$ is a basis for $V^*$ (they are linearly independent, so this part is true), and then find a basis for $V$ for which it is the dual basis. The textbook does a horrible job as explaining dual bases in general. Can someone explain me the methods behind formulating the dual basis here?

Thanks.

Solution 1:

You need to find vectors

$$ e_1 = (x_1,y_1,z_1), e_2 = (x_2,y_2,z_2), e_3 = (x_3,y_3,z_3), $$

so that

$$ f_i(e_j) = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases} $$

Write down what this means for $e_1$:

$$ x_1 - 2y_1 = 1 \\ x_1 + y_1 + z_1 = 0 \\ y_1 - 3z_1 = 0 $$

and solve for $x_1,y_1,z_1$. Then do the same for $e_2$ and $e_3$.