I'm struggling with this linear algebra 4 part question. I believe I've managed to answer part (a) of the question by applying φ to the basis elements:
I worked out $\varphi(1)=0$, $\varphi(x)=2+2x$, $\varphi(x^2)=4x+6x^2$ and $\varphi(x^3)=6x^2+12x^3$ (hope that's correct) and by combining all the columns taken from the results of the transformations, I found that:
$[\phi]^{\beta}_{\beta} = \left[\begin{array}{rrrr} 0 & 2 & 0 & 0 \\ 0 & 2 & 4 & 0 \\ 0 & 0 & 6 & 6 \\ 0 & 0 & 0 & 12 \end{array}\right] $
Not sure this is correct but for (b) does $[id]^{\beta}_{\gamma}$ just take the columns of the basis vector of Y? So $[id]^{\beta}_{\gamma}=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 0 & 1 & 3 & 6 \\ 0 & 0 & 3 & 15 \\ 0 & 0 & 0 & 15 \end{array}\right] $
but I doubt this is correct because the basis has addition signs. Any ideas on what I should do to calculate part (b)? (I think once $[id]^{\beta}_{\gamma}$ is calculated $[id]^{\gamma}_{\beta}$ is just the inverse)
Also I'm completely clueless on what to do for part (c) and (d). Please help me!
Thanks in Advance
....................................................................................................................................................................