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\begin{document}
\chead{HW Section 5.9- \#2}
\rhead{Robert Guinn\\ Math 311}


Let $T:P_{3} \rightarrow P_{4}$ be defined by $T(P) = (x + 2)p(x)$ and let $B = \{1,x,x^{2},x^{3}\}$ and $C = \{1,x,x^{2},x^{3},x^{4}\}$ be the natural bases for $P_{3}$ and $P_{4}$, respectively. Note the following:
\[
 \begin{matrix}
  T(1) = x + 2\\ 
  T(x) = x^{2} + 2x\\
  T(x^{2}) = x^{3} + 2x^{2}\\
  T(x^{3}) = x^{4} + 2x^{3}
 \end{matrix}
\]
So the coordinate vectors for $T(1)$, $T(x)$, $T(x^{2})$, and $T(x^{3})$ are
\[
 \left[T\left(1\right)\right]_{C} \,=\, \begin{bmatrix} 2\\ 1\\ 0\\ 0\\ 0 \end{bmatrix},\; \left[T\left(x\right)\right]_{C} \,=\, \begin{bmatrix} 0\\ 2\\ 1\\ 0\\ 0 \end{bmatrix},\; \left[T\left(x^{2}\right)\right]_{C} \,=\, \begin{bmatrix} 0\\ 0\\ 2\\ 1\\ 0 \end{bmatrix},\; \left[T\left(x^{3}\right)\right]_{C} \,=\, \begin{bmatrix} 0\\ 0\\ 0\\ 2\\ 1 \end{bmatrix}
\]
So the matrix representation for T is
\[
 \begin{bmatrix}
  2 & 0 & 0 & 0\\
  1 & 2 & 0 & 0\\
  0 & 1 & 2 & 0\\
  0 & 0 & 1 & 2\\
  0 & 0 & 0 & 1
 \end{bmatrix}
\]


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