Answer: The required matrix A is
[tex]A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .[/tex]
Step-by-step explanation: We are given the following linear transformation :
[tex]T:P^2\Rightarrow P^4,~~~~~~~~~~~~~T(p)=4x^2p.[/tex]
We are to find the matrix A relative to the bases [tex]B=\{1,x,x^2\}[/tex] and [tex]B^\prime=\{1,x,x^2,x^3,x^4\}.[/tex]
We have
[tex]T(1)=4x^2\times1=4x^2=0\times1+0\times x+4\times x^2+0\times x^3+0\times x^4,\\\\T(x)=4x^2\times x=4x^3=0\times1+0\times x+0\times x^2+4\times x^3+0\times x^4,\\\\T(x)=4x^2\times x^2=4x^4=0\times1+0\times x+0\times x^2+0\times x^3+4\times x^4.[/tex]
Therefore, the matrix A is of order 5 × 3, given by
[tex]A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .[/tex]
Thus, the required matrix A is
[tex]A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .[/tex]
