// Numbas version: finer_feedback_settings
{"name": "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": [{"variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5 except 0)", "name": "a", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5 except 0)", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "c", "b"], "rulesets": {}, "name": "Represent a linear map as a matrix given a basis", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "2", "maxValue": "2", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{a+c}", "maxValue": "{a+c}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{a^2+c^2}", "maxValue": "{a^2+c^2}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{a^3+c^3}", "maxValue": "{a^3+c^3}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{a^4+c^4}", "maxValue": "{a^4+c^4}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{b}", "maxValue": "{b}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{2*b*c}", "maxValue": "{2*b*c}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{3*b*c^2}", "maxValue": "{3*b*c^2}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{4*b*c^3}", "maxValue": "{4*b*c^3}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{b^2}", "maxValue": "{b^2}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{3*b^2*c}", "maxValue": "{3*b^2*c}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{6*b^2*c^2}", "maxValue": "{6*b^2*c^2}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{b^3}", "maxValue": "{b^3}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{4*b^3*c}", "maxValue": "{4*b^3*c}", "marks": 0.6}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{b^4}", "maxValue": "{b^4}", "marks": 0.6}], "type": "gapfill", "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
", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "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 $\\phi$.
", "tags": ["basis", "checked2015", "linear map", "linear spaces", "MAS2223", "matrix", "matrix given by a basis", "matrix of a linear map", "poynomials", "vector spaces"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/02/2013:
\n First draft finished.
", "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 $\\phi$.
"}, "advice": "We have:
\n\\[\\phi(1)=\\simplify[all,!zerofactor,!collectNumbers,!constantsfirst,!noleadingminus]{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[all,!zerofactor,!collectNumbers,!constantsfirst,!noleadingminus]{{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[all,!zerofactor,!collectNumbers,!constantsfirst,!noleadingminus]{ {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
\ngives the third column of the matrix.
\nContinuing on in this way for $\\phi(x^3),\\;\\phi(x^4)$ we 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}\\]
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}