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*3) .`

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*1=4x^2=0*1+0* x+4* x^2+0* x^3+0* x^4,\\\\T(x)=4x^2* x=4x^3=0*1+0* x+0* x^2+4* x^3+0* x^4,\\\\T(x)=4x^2* x^2=4x^4=0*1+0* x+0* x^2+0* x^3+4* 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*3) .`

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*3) .`