I have to express the class of the polynomial $3x^2 + 2x - 2$ in the basis of the quotient space of the real-variable polynomials of degree equal or less to 3 by the kernel of a given linear mapping.
I found that the basis of this space is $B = ((1,0,0) + ker f, (0,0,1) + ker f )$.
How do I found the coordinates of the class of the polynomial in this basis?
(I assume you meant $\mathbb{R}[x]_{\leq2}$ instead of $\mathbb{R}[x]_{\leq3}$.)
The coordinates of the polynomial $3x^2+2x-2$ in the canonical basis of $\mathbb{R}[x]_{\leq2}$, $\{1,X,X^2\}$ are $(-2,2,3)$.
The basis $B=((1,0,0)+kerf,(0,0,1)+kerf)$ has been obtained by completion of a basis of $kerf$, so I will asume $(0,1,0)$ is such base.
Since the quotient space is a vector space we can express the class of $(-2,2,3)$,$[(-2,2,3)]=[(-2,0,0)]+[(0,2,0)]+[(0,0,3)]=-2[(1,0,0)]+2[(0,1,0)]+3[(0,0,1)]$. Since (0,1,0) is basis of $kerf$ then $[(0,1,0)]=[0]$.
Therefore $[(-2,2,3)]=(-2,3)$ in basis $B$.
I hope it helped.