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{"name": "Simon's copy of Represent a linear map as a matrix given a basis", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Simon's copy of Represent a linear map as a matrix given a basis", "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["a", "c", "b"], "statement": "Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ with coefficients in the real numbers.
\n Let the linear map $\\phi: P_4 \\rightarrow P_4$ be defined by:
\n \\[\\phi(p(x))=p(\\var{a})+p(\\simplify{{b}x+{c}})\\]
\n 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 $p(x)$.
\n For example:
\n $\\phi(x^2+2x)=\\simplify[all,!collectnumbers,!noleadingminus]{{a}^2+2*{a}+({b}x+{c})^2+2*({b}x+{c})={a^2+2*a+2*c}+{2*b*c+2*b}*x+{b^2}*x^2}$.
\n Using the standard basis for range and domain find the matrix given by $\\phi$.
", "parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "prompt": "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 | \\[\\left( \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n [[0]] | \n [[1]] | \n [[2]] | \n [[3]] | \n [[4]] | \n \\[\\left) \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\\] | \n
\n \n | $0$ | \n [[5]] | \n [[6]] | \n [[7]] | \n [[8]] | \n
\n \n | $0$ | \n $0$ | \n [[9]] | \n [[10]] | \n [[11]] | \n
\n \n | $0$ | \n $0$ | \n $0$ | \n [[12]] | \n [[13]] | \n
\n \n | $0$ | \n $0$ | \n $0$ | \n $0$ | \n [[14]] | \n
\n \n
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"variableReplacements": [], "showFeedbackIcon": true}, {"type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "{b^4}", "useCustomName": false, "showFractionHint": true, "extendBaseMarkingAlgorithm": true, "customName": "", "customMarkingAlgorithm": "", "allowFractions": false, "minValue": "{b^4}", "mustBeReduced": false, "mustBeReducedPC": 0, "marks": 0.6, "unitTests": [], "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true}], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true}], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "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 $\\phi$.
"}, "tags": [], "functions": {}, "advice": "We have:
\n\\[\\phi(1)=\\simplify[]{1+1= 2*1= 2 * 1 + 0 * x + 0 * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}\\]
\ngives the first column of the matrix.
\n\\[\\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}\\]
\ngives the second column of the matrix.
\n\\[\\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}\\]
\n gives the third column of the matrix.
\n\nContinuing on in this way for $\\phi(x^3),\\;\\phi(x^4)$:
\n\\[\\phi(x^3)=\\simplify[]{ {a}^ 3 + ({b} * x + {c})^3}= ... \\]
\n\\[\\phi(x^4)=\\simplify[]{ {a}^ 4 + ({b} * x + {c})^4}= ... \\]
\nwe obtain the matrix for $\\phi$ with respect to the given bases for domain and range.
\n
\n\\[\\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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