Let T:P2→P2 be the linear operator defined by T(p(x))=p(2x+1). Find the matrix of the operator T with respect to the bases S={p1,p2,p3}, where p1=1,p2=x,p3=x². Also, compute T(2-3x+4x²).

Question

Let [latex]T\colon P_{2} o P_{2}[/latex] be the linear operator defined by T(p(x))=p(2x+1). Find the matrix of the operator T with respect to the bases [latex]S=\left\{p_{1},p_{2},p_{3} ight\},[/latex] where [latex]p_{1}=1,p_{2}=x,p_{3}=x^{2}.[/latex] Also, compute [latex]T(2-3x+4x^{2}).[/latex]

Accepted Answer

Here, [latex]T(p_{1})=T(1)=1[/latex]

[latex] {{T(p_{2})=T(x)=2x+1,}}\[0.5 em] {{T(p_{3})=T\left(x^{2}\right)=\left(2x+1\right)^{2}=4x^{2}+4x+1.}}[/latex]

Let us express [latex]p_{1}[/latex] as a linear combination of [latex]p_{1},p_{2},p_{3}[/latex]

[latex] \begin{array}{c}{{T(p_{1})=k_{1}p_{1}+k_{2}p_{2}+k_{3}p_{3}}}\[0.5 em] \therefore 1=k_{1}1+k_{2}x+k_{3}x^{2}. \end{array}[/latex]

Comparing both sides, coefficient of [latex]1,x,x^{2}[/latex]

[latex] \left.\begin{array}{c}{{k_{1}=1,k_{2}=0,k_{3}=0}}\[0.5 em] {{T(p_{1})=1}}\[0.5 em] {\pmb{\big[}T(p_{1})\pmb{\big]} _{s}}= {{\left[\ \begin{array}{c}{{1}}\ {{0}}\ {{0}}\end{array}\ \right].}}\end{array}\right.[/latex]

Express [latex]p_{2}[/latex] as a linear combination of [latex]p_{1},p_{2},p_{3}[/latex]

[latex] \begin{array}{c}{{T(p_{2})=k_{1}p_{1}+k_{2}p_{2}+k_{3}p_{3}}}\[0.5 em] \therefore 2x+1=k_{1}1+k_{2}x+k_{3}x^{2}. \end{array}[/latex]

Comparing both sides, coefficient of [latex]1,x,x^{2}[/latex]

[latex] \left.\begin{array}{c}{{k_{1}=1,k_{2}=2,k_{3}=0}}\[0.5 em] T(p_{2})=p_1+2p_2 \[0.5 em] {\pmb{\big[}T(p_{2})\pmb{\big]} _{s}}= {{\left[\ \begin{array}{c}{{1}}\ {{2}}\ {{0}}\end{array}\ \right].}}\end{array}\right.[/latex]

Express [latex]p_{3}[/latex] as a linear combination of [latex]p_{1},p_{2},p_{3}[/latex]

[latex]\begin{array}{c}{{T(p_{3})=k_{1}p_{1}+k_{2}p_{2}+k_{3}p_{3}}}\[0.5 em] \therefore 4x^2+4x+1=k_{1}1+k_{2}x+k_{3}x^{2}. \end{array}[/latex]

Comparing both sides, coefficient of [latex]1,x,x^{2}[/latex]

[latex]\left.\begin{array}{c}{{k_{1}=1,k_{2}=4,k_{3}=4}}\[0.5 em] T(p_{2})=p_1+4p_2+4p_3 \[0.5 em] {\pmb{\big[}T(p_{3})\pmb{\big]} _{s}}= {{\left[\ \begin{array}{c}{{1}}\ {{4}}\ {{4}}\end{array}\ \right].}}\end{array}\right.[/latex]

The matrix of the operator T with respect to the basis S is

[latex]\pmb{\big[}T\pmb{\big]}_s =\left[\pmb{\big[}T(p_{1})\pmb{\big]}_{s}|\pmb{\big[}T(p_{2})\pmb{\big]}_{s}|\pmb{\big[}T(p_{3})\pmb{\big]}_{s}\right]=\left[\ {\begin{array}{c c c}{1}&{1}&1\ {0}&{2}&4\ {0}&{0}&4\end{array}}\ \right].[/latex]

Question 4.2: Let T:P2→P2 be the linear operator defined by T(p(x))=p(2x+1......

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