// Numbas version: finer_feedback_settings {"name": "Week 9 (non-assessed)", "metadata": {"description": "", "licence": "All rights reserved"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "vectors in R^2a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Charles Garnet Cox", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24585/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

These questions involve the bases $A=\\left\\{\\begin{pmatrix}1\\\\1\\end{pmatrix}, \\begin{pmatrix}1\\\\-1\\end{pmatrix}\\right\\}$ and $E=\\left\\{\\begin{pmatrix}1\\\\0\\end{pmatrix}, \\begin{pmatrix}0\\\\1\\end{pmatrix}\\right\\}$.

", "advice": "

In all parts, we need to understand what the subscript on the vector means.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "2+n", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "2-n", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "0.5*(d1+d2)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "0.5*(d1-d2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k", "m", "n", "c1", "c2", "d1", "d2", "d", "e"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find $a\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}\\var{k}\\\\0\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{k}\\\\a\\end{pmatrix}_{E}$

a=[[0]]

Find $b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}\\var{m}\\\\\\var{m}\\end{pmatrix}_{A}=\\begin{pmatrix}b\\\\0\\end{pmatrix}_{E}$

b=[[0]]

Find $c\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}2\\\\c\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{c1}\\\\\\var{c2}\\end{pmatrix}_{E}$

c=[[0]]

Find $a, b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}a\\\\b\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{d1}\\\\\\var{d2}\\end{pmatrix}_{E}$

a=[[0]], b=[[1]]

", "advice": "

In all parts, we need to understand what the subscript on the vector means.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "2+n", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "2-n", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "-m", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "0.5*(d1+d2)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "0.5*(d1-d2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k", "m", "n", "c1", "c2", "m2", "d1", "d2", "d", "e"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find $a\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}0\\\\\\var{k}\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{k}\\\\a\\end{pmatrix}_{E}$

a=[[0]]

Find $b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}\\var{m}\\\\\\var{m2}\\end{pmatrix}_{A}=\\begin{pmatrix}0\\\\b\\end{pmatrix}_{E}$

b=[[0]]

Find $c\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}2\\\\c\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{c1}\\\\\\var{c2}\\end{pmatrix}_{E}$

c=[[0]]

Find $a, b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}a\\\\b\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{d1}\\\\\\var{d2}\\end{pmatrix}_{E}$

a=[[0]], b=[[1]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "d", "maxValue": "d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "e", "maxValue": "e", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "change of basis simpler general in R^2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Charles Garnet Cox", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24585/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

These questions involve the bases $A=\\left\\{\\begin{pmatrix}\\var{a}\\\\\\var{c}\\end{pmatrix}, \\begin{pmatrix}\\var{b}\\\\\\var{d}\\end{pmatrix}\\right\\}$ and $E=\\left\\{\\begin{pmatrix}1\\\\0\\end{pmatrix}, \\begin{pmatrix}0\\\\1\\end{pmatrix}\\right\\}$.

(If your answer is best expressed as a fraction, you can do this by typing an expression of the form 'x/y' where x and y are integers.)

", "advice": "

Consider what size the matrices $C_{EA}$ and $C_{AE}$ should have. Recall that $C_{EA}$ converts vectors from 'the language' of A to 'the language' of E.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "d+1", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a-1", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "a", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(a*d)/c", "description": "", "templateType": "anything", "can_override": false}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "1/(a*d-b*c)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "disc*(d*c1-b*c2)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "disc*(-c*c1+a*c2)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "e2": {"name": "e2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c1", "c2", "a", "c", "b", "d", "x", "disc", "m", "n", "d1", "d2", "e1", "e2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find $a, b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}a\\\\b\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{c1}\\\\\\var{c2}\\end{pmatrix}_{E}$

a=[[0]], b=[[1]]

Find the following matrix.

$C_{EA}=$[[0]]

Hence find

$\\begin{pmatrix}\\var{d1}\\\\\\var{d2}\\end{pmatrix}_{A}=$[[1]]$_E$

", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([a,b],[c,d])", "correctAnswerFractions": true, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([a*d1+b*d2],[c*d1+d*d2])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the following matrix.

$C_{AE}=$[[0]]

Hence find

$\\begin{pmatrix}\\var{e1}\\\\\\var{e2}\\end{pmatrix}_{E}=$[[1]]$_A$

", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([disc*d,-disc*b],[-disc*c,disc*a])", "correctAnswerFractions": true, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([disc*d*e1-disc*b*e2],[-disc*c*e1+disc*a*e2])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "change of basis general in R^2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Charles Garnet Cox", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24585/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

(If your answer is best expressed as a fraction, you can do this by typing an expression of the form 'x/y' where x and y are integers.)

", "advice": "

Consider what size the matrices $C_{EA}$ and $C_{AE}$ should have. Recall that $C_{EA}$ converts vectors from 'the language' of A to 'the language' of E.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-2..4 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-2..4 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "d+1", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a-1", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-2..2 except a+1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(a*d)/c", "description": "", "templateType": "anything", "can_override": false}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "1/(a*d-b*c)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "disc*(d*c1-b*c2)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "disc*(-c*c1+a*c2)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "e2": {"name": "e2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c1", "c2", "a", "c", "b", "d", "x", "disc", "m", "n", "d1", "d2", "e1", "e2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find $a, b\\in\\mathbb{R}$ such that the following are equal.

$\\begin{pmatrix}a\\\\b\\end{pmatrix}_{A}=\\begin{pmatrix}\\var{c1}\\\\\\var{c2}\\end{pmatrix}_{E}$

a=[[0]], b=[[1]]

Find the following matrix.

$C_{EA}=$[[0]]

Hence find

$\\begin{pmatrix}\\var{d1}\\\\\\var{d2}\\end{pmatrix}_{A}=$[[1]]$_E$

Find the following matrix.

$C_{AE}=$[[0]]

Hence find

$\\begin{pmatrix}\\var{e1}\\\\\\var{e2}\\end{pmatrix}_{E}=$[[1]]$_A$

", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([disc*d,-disc*b],[-disc*c,disc*a])", "correctAnswerFractions": true, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([disc*d*e1-disc*b*e2],[-disc*c*e1+disc*a*e2])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "change of basis in R^3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Charles Garnet Cox", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24585/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

These questions involve the bases $B=\\left\\{\\begin{pmatrix}2\\\\0\\\\0\\end{pmatrix}, \\begin{pmatrix}1\\\\1\\\\0\\end{pmatrix}, \\begin{pmatrix}0\\\\1\\\\1\\end{pmatrix}\\right\\}$ and $E=\\left\\{\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}, \\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}, \\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}\\right\\}$.

", "advice": "

Understanding of the subscripts on the given vectors will allow us to set up simultaneous linear equations that we can solve.

Find $a, b, c\\in\\mathbb{R}$ such that the following are equal. (Your answers may be fractions.) $\\begin{pmatrix}\\var{c1}\\\\\\var{c2}\\\\\\var{c3}\\end{pmatrix}_{E}=\\begin{pmatrix}a\\\\b\\\\c\\end{pmatrix}_{B}$

a=[[0]], b=[[1]], c=[[2]]

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Throughout this question, we work with

$A=M_{EE}(f)=\\begin{pmatrix}1&1&2\\\\2&2&4\\\\3&3&6\\end{pmatrix}.$

Note that $E$ denotes the standard basis and $f: \\mathbb{R}^3\\to \\mathbb{R}^3$.

(This matrix does not change if you refresh the question.)

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For parts (a)-(c), it could be worth looking over the related exercises on diagonalisation that appear on the corresponding problem sheet.

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The vector $w=\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix}$ is an eigenvector corresponding to the eigenvalue [[0]].

Consider whether our matrix has a non-trivial kernel.

Can you find a vector $v$ in $\\ker(A)$? [[0]]

Are scalar multiples of $v$, your vector, in $\\ker(A)$? [[1]]

Can you find another vector in $\\ker(A)$ which is not a scalar multiple of $v$? [[2]]

What is $\\dim(\\ker(A))$? [[3]]

Before moving on to the next part, it is recommended that you find a basis of eigenvectors for our matrix $A$.

Note that if $v, w \\in \\ker(A)$, then any element of $\\textrm{span}\\{v, w\\}$ is also in $\\ker(A)$.

Let $v_1$ and $v_2$ be defined as follows.

$v_1=\\begin{pmatrix}1\\\\-1\\\\0\\end{pmatrix}$ and $v_2=\\begin{pmatrix}2\\\\0\\\\-1\\end{pmatrix}$.

Check that $v_1, v_2 \\in \\ker(A)$. [[0]]

Check that your eigenvectors (calculated in (b) lie in $\\textrm{span}\\{v_1, v_2\\}$. [[1]]

Using the vector $w$ from part (a), we have that $B=\\{v_1, v_2, w\\}$ is a basis of eigenvectors. [[2]]

We can use the result from lectures to find $M_{BB}(f)$, which will be a diagonal matrix. [[3]]

We continue with $B=\\{v_1, v_2, w\\}$ where $v_1=\\begin{pmatrix}1\\\\-1\\\\0\\end{pmatrix}$ and $v_2=\\begin{pmatrix}2\\\\0\\\\-1\\end{pmatrix}$ and $w=\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix}$.

Consider each of the following sets of vectors, and select answers from the drop-down lists.

$\\{v_1, 2v_1, w\\}$ [[0]]
$\\{v_2, 2v_1, w\\}$ [[1]]
$\\{w, v_1, v_2\\}$ [[2]]
$\\{v_1, v_2, v_1+v_2, w\\}$ [[3]]
$\\{v_1, v_2, w+v_1\\}$ [[4]]
$\\{v_1, {\\bf 0}, w\\}$ [[5]]
$\\{v_1+v_2, v_1-v_2, w\\}$ [[6]]
$E$, the standard basis. [[7]]

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This is a non-assessed quiz for MATH10015 Linear Algebra.

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