Question

LetT: P2 → P4be the linear transformationT(p) = 4x2p.Find the matrix A for T relative to the basesB = {1, x, x2}andB' = {1, x, x2, x3, x4}.

Answer 1

Answer:  The required matrix A is

$A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .$

Step-by-step explanation:  We are given the following linear transformation :

$T:P^2\Rightarrow P^4,~~~~~~~~~~~~~T(p)=4x^2p.$

We are to find the matrix A relative to the bases $B=\{1,x,x^2\}$ and $B^\prime=\{1,x,x^2,x^3,x^4\}.$

We have

$T(1)=4x^2\times1=4x^2=0\times1+0\times x+4\times x^2+0\times x^3+0\times x^4,\\\\T(x)=4x^2\times x=4x^3=0\times1+0\times x+0\times x^2+4\times x^3+0\times x^4,\\\\T(x)=4x^2\times x^2=4x^4=0\times1+0\times x+0\times x^2+0\times x^3+4\times x^4.$

Therefore, the matrix A is of order 5 × 3, given by

$A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .$

Thus, the required matrix A is

$A=\left[\begin{array}{ccc}0&0&0\\0&0&0\\4&0&0\\0&4&0\\0&0&4\end{array}\right]_{5\times3} .$