// Numbas version: finer_feedback_settings
{"name": "John'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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "advice": "<p>We have:</p>\n<p>\\[\\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}\\]</p>\n<p>gives the first column of the matrix.</p>\n<p>\\[\\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}\\]</p>\n<p>gives the second column of the matrix.</p>\n<p>\\[\\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}\\]</p>\n<p>&nbsp;</p>\n<p>gives the third column of the matrix.</p>\n<p>Continuing 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.</p>\n<p>&nbsp;</p>\n<p>\\[\\begin{pmatrix}\\var{2}&amp;\\var{a+c}&amp;\\var{a^2+c^2}&amp;\\var{a^3+c^3}&amp;\\var{a^4+c^4}\\\\0&amp;\\var{b}&amp;\\var{2*b*c}&amp;\\var{3*b*c^2}&amp;\\var{4*b*c^3}\\\\0&amp;0&amp;\\var{b^2}&amp;\\var{3*b^2*c}&amp;\\var{6*b^2*c^2}\\\\0&amp;0&amp;0&amp;\\var{b^3}&amp;\\var{4*b^3*c}\\\\0&amp;0&amp;0&amp;0&amp;\\var{b^4}\\end{pmatrix}\\]</p>", "ungrouped_variables": ["a", "c", "b"], "parts": [{"prompt": "<p>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:</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"5\">\\[\\left( \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td rowspan=\"5\">\\[\\left) \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>[[7]]</td>\n        <td>[[8]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[9]]</td>\n        <td>[[10]]</td>\n        <td>[[11]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[12]]</td>\n        <td>[[13]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[14]]</td>\n        </tr>\n        </tbody>\n        </table>", "showCorrectAnswer": true, "gaps": [{"allowFractions": false, "showCorrectAnswer": true, "minValue": "2", "maxValue": "2", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{a+c}", "maxValue": "{a+c}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{a^2+c^2}", "maxValue": "{a^2+c^2}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{a^3+c^3}", "maxValue": "{a^3+c^3}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{a^4+c^4}", "maxValue": "{a^4+c^4}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{b}", "maxValue": "{b}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{2*b*c}", "maxValue": "{2*b*c}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{3*b*c^2}", "maxValue": "{3*b*c^2}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{4*b*c^3}", "maxValue": "{4*b*c^3}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{b^2}", "maxValue": "{b^2}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{3*b^2*c}", "maxValue": "{3*b^2*c}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{6*b^2*c^2}", "maxValue": "{6*b^2*c^2}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{b^3}", "maxValue": "{b^3}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{4*b^3*c}", "maxValue": "{4*b^3*c}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}, {"allowFractions": false, "showCorrectAnswer": true, "minValue": "{b^4}", "maxValue": "{b^4}", "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "marks": 0.6, "showPrecisionHint": false}], "scripts": {}, "marks": 0, "type": "gapfill"}], "preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "<p><strong>15/02/2013:</strong></p>\n    <p>First draft finished.</p>", "description": "<p>Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ &nbsp;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 $\\phi$.</p>"}, "variable_groups": [], "tags": ["basis", "checked2015", "linear map", "linear spaces", "MAS2223", "matrix", "matrix given by a basis", "matrix of a linear map", "poynomials", "vector spaces"], "type": "question", "functions": {}, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "name": "", "questions": []}], "showQuestionGroupNames": false, "variables": {"b": {"definition": "random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b"}, "a": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a"}, "c": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "c"}}, "name": "John's copy of Represent a linear map as a matrix given a basis", "statement": "<p>Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ &nbsp;with coefficients in the real numbers.</p>\n  <p>Let the linear map $\\phi: P_4 \\rightarrow P_4$ be defined by:</p>\n  <p>\\[\\phi(p(x))=p(\\var{a})+p(\\simplify{{b}x+{c}})\\]</p>\n  <p>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 $p(x)$.</p>\n  <p>For example:</p>\n  <p>$\\phi(x^2+2x)=\\simplify[all,!collectnumbers,!noleadingminus]{{a}^2+2*{a}+({b}x+{c})^2+2*({b}x+{c})={a^2+2*a+2*c}+{2*b*c+2*b}*x+{b^2}*x^2}$.</p>\n  <p>Using the standard basis for range and domain find the matrix given by $\\phi$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}