Finding coordinates with respect to a basis

Question

Let $B={{(1,x,x^2)}}$ and $C=(1,3x+4x^2,2x+3x^2)$ be bases for $P_2(\mathbb R)$.

Find the coordinates of $x$ and $x^2$ with respect to the basis $C$.

I'm a little stuck on where do begin for this question, any help is appreciated, thanks!

Answer 1

As $C$ is a basis for $P_2(\mathbb{R})$, then you may write $x$ in terms of the vectors of basis $C$. This being said, there exist $c_1,c_2,c_3 \in \mathbb{R}$ such that

$$x = c_1 + c_2 (3x+4x^2) + c_3(2x+3x^2) = c_1 + (3c_2+2c_3)x + (4c_2 + 3c_3)x^2$$

Then solve

$$\begin{align}\begin{cases}c_1 &=0\\3c_2+2c_3 &= 1\\ 4c_2 + 3c_3 &=0\end{cases} \end{align}$$

to find $c_1,c_2,c_3$. Similar approach to $x^2$.