# Consider the following linear transformation T : P_2 rightarrow

Consider the following linear transformation $T:{P}_{2}\to {P}_{3}$, given by $T\left(f\right)=3{x}^{2}f$'.

That is, take the first derivative and then multiply by $3{x}^{2}$

(a) Find the matrix for T with respect to the standard bases of ${P}_{n}$: that is, find $\left[T{\right]}_{ϵ}^{ϵ}$, where- $ϵ=1,x,{x}^{2},{x}^{n}$

(b) Find N(T) and R(T). You can either work with polynomials or with their coordinate vectors with respect to the standard basis. Write the answers as spans of polynomials.

(c) Find the the matrix for T with respect to the alternate bases: $\left[T{\right]}_{A}^{B}$ where $A=x-1,x,{x}^{2}+1,B={x}^{3},x,{x}^{2},1.$

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Sulotion: Given that nebce that to the $T:{P}_{2}\to {P}_{3}T\left(f\right)=3{x}^{2}{f}^{\prime }$ a)

$\left[T{\right]}_{B}^{\gamma }=A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 3& 0\\ 0& 0& 6\end{array}\right]$

b. That to that the

Let then the $n\left(1\right)=5pon|\left\{1,0,x\right\}$ spon let $P\left(x\right)=7+8x+\gamma {x}^{2}\in {P}_{2}$ hence that to

Let then to $T\left(P\right)=A-\left[\begin{array}{c}x\\ B\\ \gamma \end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 3& 0\\ 0& 0& 6\end{array}\right]\left[\begin{array}{c}x\\ 0B\\ \gamma \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 3B\\ 6\gamma \end{array}\right]$

Then the $R\left(T\right)=\left\{\left(0,0,1,0{\right)}^{T}\left(0,0,0{\right)}^{T}\right\}$ is spon $\left\{{x}^{2},x\right\}$ c. $A=\left\{{x}^{-1},x,{x}^{2}+1\right\}B=\left\{{x}^{3},x,{x}^{2}\right\}$ Let the then $T\left(x-1\right)=3{x}^{2}-1=0.{x}^{3}+0.x+3{x}^{2}+0.1$
$T\left(x\right)=3{x}^{2}-\left(1\right)=0.{x}^{3}+0.1$
$T\left({x}^{2}+1\right)=3{x}^{2}\left(2x\right)=6.{x}^{3}+0.x+0.{x}^{2}+0.1$
$\left[T{\right]}_{A}^{B}=\left[\begin{array}{ccc}0& 0& 6\\ 0& 0& 0\\ 3& 3& 0\\ 0& 0& 0\end{array}\right]$