Have the following let $P_2(R)$ denote a vector space of the real polynomial functions of degree less than or equal to two and let $B:=[P_0,P_1,P_2]$ denote the natural ordered basis for $P_2(R)$ (so $P_i(x)=x^{i})$ . Define $f\in P_2(R)$ by $f(x)=5x^{2}-2x+3$ . Write f as a linear combination of the elements of B. Compute the coordinate vector $f_B$ of f with repect to B.
$f:= 5p_2-2p_1+3p_0$ $\rightarrow$ $f_B:=[3,-2,5]$
Define $h_1,h_2,h_3\in P_2(R)$ by $h_1(x)= 7x^{2}+3x+4$, $h_2(x)= x^{2}+2x+1$, $h_3(x)= x^{2}-1$. Define $C:=[h_1,h_2,h_3]$. Assume C is an ordered basis for $P_2(R)$, construct the change of coordinate matrix, A, which converts C-coordinate to B-coordinates. Compute $A^{-1}$.
$A= \begin{pmatrix}4 & 1 & -1\\3 & 2 & 0\\7 &1 &1\\\end{pmatrix}$ $\Rightarrow$ $A^{-1}$= $\begin{pmatrix}2/16 & -2/16 & 2/16\\-3/16 & 11/16 & -3/16\\-11/16 &3/16 &5/16\\\end{pmatrix}$
Im struggling with the following Let F: $P_2(R)$ $\rightarrow$ $P_3(R)$ be the linear transformation determined by: $F(f)(x)=6\int_{0}^{2x-1} f(t) dt$
Compute the dimension of the kernel of F. Determine a basis for the image of F. Define $A:=[P_0,P_1,P_2,P_3]$ and compute $M_{B}^{A}(F)$, the matrix of F with respect to the given ordered bases. Determine the rank of $M_{C}^{A}(F)$.
Many thanks in advance.