Mathematics: 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}.

Lett: p2 → p4be the linear transformationt(p) =

Ответ:

60129

12.01.2022

Mathematics

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

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

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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}.

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