// Numbas version: finer_feedback_settings
{"name": "Represent a linear map as a matrix with a given 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}, "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "a"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "f"}}, "ungrouped_variables": ["a", "c", "b", "d", "f"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Represent a linear map as a matrix with a given basis", "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a}", "minValue": "{a}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c}", "minValue": "{c}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{2*f}", "minValue": "{2*f}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+b}", "minValue": "{a+b}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{2*d+2*c}", "minValue": "{2*d+2*c}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{6*f}", "minValue": "{6*f}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+2*b+2}", "minValue": "{a+2*b+2}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{3*c+6*d}", "minValue": "{3*c+6*d}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{12*f}", "minValue": "{12*f}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+3*b+6}", "minValue": "{a+3*b+6}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{4*c+12*d}", "minValue": "{4*c+12*d}", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "showCorrectAnswer": true, "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+4*b+12}", "minValue": "{a+4*b+12}", "showCorrectAnswer": true, "marks": 0.5}], "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.\n        \n        Fill in the entries for this matrix below: \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>$0$</td>\n      <td>$0$</td>\n      <td rowspan=\"5\">\\[\\left) \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\\]</td>\n      </tr><tr>\n      <td>[[3]]</td>\n      <td>[[4]]</td>\n      <td>[[5]]</td>\n      <td>[[6]]</td>\n      <td>$0$</td>\n      </tr>\n      <tr>\n      <td>$0$</td>\n      <td>[[7]]</td>\n      <td>[[8]]</td>\n      <td>[[9]]</td>\n      <td>[[10]]</td>\n      </tr>\n      <tr>\n      <td>$0$</td>\n      <td>$0$</td>\n      <td>[[11]]</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>[[14]]</td>\n      <td>[[15]]</td>\n      </tr>\n      </tbody>\n      </table>", "showCorrectAnswer": true, "marks": 0}], "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)) = \\simplify[all,!collectnumbers]{{a} * p(x) + ({b} * x + {c}) * p'(x) + (x ^ 2 + {d} * x + {f}) * p''(x)}\\]</p>\n<p>where &nbsp;$p'(x)$ is the first derivative of $p(x)$ &nbsp;and $p''(x)$ the second derivative.&nbsp;</p>", "tags": ["basis", "checked2015", "linear map", "linear spaces", "MAS2223", "matrix", "matrix given by a basis", "matrix of a linear map", "poynomials", "vector spaces"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>13/02/2013:</strong></p>\n                        <p>Advice to be completed.</p>", "licence": "Creative Commons Attribution 4.0 International", "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.</p>\n                <p>Let the linear map $\\phi: P_4 \\rightarrow P_4$ be defined by:</p>\n                <p>$\\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$</p>\n                <p>Using the standard basis for range and domain find the matrix given by $\\phi$.</p>"}, "advice": "<p>We have:</p>\n<p>\\[\\phi(1) =\\simplify[all,!zerofactor,!collectNumbers,!constantsfirst,!noleadingminus]{ {a} * 1 + ({b} * x + {c}) * 0 + (x ^ 2 + {d} * x + {f}) * 0 = {a} = {a} * 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} * x + ({b} * x + {c}) * 1 + (x ^ 2 + {d} * x + {f}) * 0 = {c} + {a + b} * x = {c} * 1 + {a + 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} * x ^ 2 + ({b} * x + {c}) * 2 * x + (x ^ 2 + {d} * x + {f}) * 2 = {2 * f} + {2 * d + 2 * c} * x + {a + 2 * b + 2} * x ^ 2 = {2 * f} * 1 + {2 * d + 2 * c} * x + {a + 2 * 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>\\[\\begin{pmatrix}\\var{a}&amp;\\var{c}&amp;\\var{2*f}&amp;0&amp;0\\\\0&amp;\\var{a+b}&amp;\\var{2*d+2*c}&amp;\\var{6*f}&amp;0\\\\0&amp;0&amp;\\var{a+2*b+2}&amp;\\var{3*c+6*d}&amp;\\var{12*f}\\\\0&amp;0&amp;0&amp;\\var{a+3*b+6}&amp;\\var{4*c+12*d}\\\\0&amp;0&amp;0&amp;0&amp;\\var{a+4*b+12}\\end{pmatrix}\\]</p>", "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/"}]}