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Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

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Let the linear map $\\phi: P_4 \\rightarrow P_4$ be defined by:

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\$\\phi(p(x))=p(\\var{a})+p(\\simplify{{b}x+{c}})\$

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This is given by evaluating $p(x)$ at $x=\\var{a}$ and adding this to the polynomial given by replacing $x$ by $\\simplify{{b}x+{c}}$ in $p(x)$.

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For example:

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$\\phi(x^2+2x)=\\simplify[all,!collectnumbers,!noleadingminus]{{a}^2+2*{a}+({b}x+{c})^2+2*({b}x+{c})={a^2+2*a+2*c}+{2*b*c+2*b}*x+{b^2}*x^2}$.

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Using the standard basis for range and domain find the matrix given by $\\phi$.

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Using the ordered basis $\\{1,\\;x,\\;x^2,\\;x^3,\\;x^4\\}$ of $P_4$ for both range and domain, $\\phi$ is represented by a 5 x 5 matrix. Fill in the entries for this matrix below:

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
 \$\\left( \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\$ [[0]] [[1]] [[2]] [[3]] [[4]] \$\\left) \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\$ $0$ [[5]] [[6]] [[7]] [[8]] $0$ $0$ [[9]] [[10]] [[11]] $0$ $0$ $0$ [[12]] [[13]] $0$ $0$ $0$ $0$ [[14]]
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Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\\phi: P_4 \\rightarrow P_4$ be defined by: \$\\phi(p(x))=p(a)+p(bx+c).\$Using the standard basis for range and domain find the matrix given by $\\phi$.

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We have:

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\$\\phi(1)=\\simplify[]{1+1= 2*1= 2 * 1 + 0 * x + 0 * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}\$

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gives the first column of the matrix.

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\$\\phi(x)=\\simplify[]{{a}+ ({b} * x + {c}) = {a+c} + {b} * x = {a+c} * 1 + {b} * x + 0 * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}\$

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gives the second column of the matrix.

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\$\\phi(x^2)=\\simplify[]{ {a}^ 2 + ({b} * x + {c})^2 = {a^2+c^2} + {2 * b* c} * x + {b^2} * x ^ 2 = {a^2+c^2} * 1 + {2*b*c} * x + {b^2} * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}\$

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gives the third column of the matrix.

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Continuing on in this way for $\\phi(x^3),\\;\\phi(x^4)$:

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\$\\phi(x^3)=\\simplify[]{ {a}^ 3 + ({b} * x + {c})^3}= ... \$

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\$\\phi(x^4)=\\simplify[]{ {a}^ 4 + ({b} * x + {c})^4}= ... \$

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we obtain the matrix for $\\phi$ with respect to the given bases for domain and range.

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\$\\begin{pmatrix}\\var{2}&\\var{a+c}&\\var{a^2+c^2}&\\var{a^3+c^3}&\\var{a^4+c^4}\\\\0&\\var{b}&\\var{2*b*c}&\\var{3*b*c^2}&\\var{4*b*c^3}\\\\0&0&\\var{b^2}&\\var{3*b^2*c}&\\var{6*b^2*c^2}\\\\0&0&0&\\var{b^3}&\\var{4*b^3*c}\\\\0&0&0&0&\\var{b^4}\\end{pmatrix}\$

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