// Numbas version: finer_feedback_settings {"name": "Week 8 (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": "Standard basis to matrix formv1", "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": "

Standard basis to matrix form

First decide on what size of matrix we should use.

(Hint: consider the domain and co-domain of f.)

(Hint 2: the size of the vectors that are inputs and outputs indicate the dimensions the matrix should have.)

We then want to use the FACT from Linear Algebra.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "l2": {"name": "l2", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "l31": {"name": "l31", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "l34": {"name": "l34", "group": "Ungrouped variables", "definition": "random(l33+1..3)", "description": "", "templateType": "anything", "can_override": false}, "l22": {"name": "l22", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "l32": {"name": "l32", "group": "Ungrouped variables", "definition": "random(l31+1..3)", "description": "", "templateType": "anything", "can_override": false}, "l33": {"name": "l33", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "i": {"name": "i", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l2", "l22", "l31", "l32", "l33", "l34"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "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": "

This question involves writing a function in standard basis form as an appropriately sized matrix.

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We have an $\\mathbb{R}$-linear function $f: \\mathbb{R}^2\\to \\mathbb{R}^1$ where

\\[f(e_1)=\\var{d}e_1\\textrm{ and }f(e_2)=\\var{e}e_1.\\]

Write $f$ in matrix form.

", "correctAnswer": "matrix([d,e])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

We have an $\\mathbb{R}$-linear function $g_1: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_1(e_1)=\\var{f}e_1\\textrm{ and }g_1(e_2)=\\var{g}e_2.\\]

Write $g_1$ in matrix form.

", "correctAnswer": "matrix([f,0],[0,g])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

We have an $\\mathbb{R}$-linear function $g_2: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_2(e_1)=\\var{k}e_2\\textrm{ and }g_2(e_2)=(\\var{i})e_2.\\]

Write $g_2$ in matrix form.

", "correctAnswer": "matrix([0,0],[k,i])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

We have an $\\mathbb{R}$-linear function $h: \\mathbb{R}^2\\to \\mathbb{R}^3$ where

\\[h(e_1)=\\var{h}e_1+(\\var{i})e_2\\textrm{ and }h(e_2)=\\var{j}e_1+(\\var{k})e_3.\\]

Write $h$ in matrix form.

", "correctAnswer": "matrix([h,j],[i,0],[0,k])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Standard basis to matrix formv2", "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": "

Standard basis to matrix form

First decide on what size of matrix we should use.

(Hint: consider the domain and co-domain of f.)

(Hint 2: the size of the vectors that are inputs and outputs indicate the dimensions the matrix should have.)

We then want to use the FACT from Linear Algebra.

This question involves writing a function in standard basis form as an appropriately sized matrix.

We have an $\\mathbb{R}$-linear function $f: \\mathbb{R}^2\\to \\mathbb{R}^1$ where

\\[f(e_1)=\\var{d}e_1\\textrm{ and }f(e_2)=\\var{e}e_1.\\]

Write $f$ in matrix form.

We have an $\\mathbb{R}$-linear function $g_1: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_1(e_1)=\\var{f}e_1\\textrm{ and }g_1(e_2)=\\var{g}e_2.\\]

Write $g_1$ in matrix form.

We have an $\\mathbb{R}$-linear function $g_2: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_2(e_1)=\\var{k}e_2\\textrm{ and }g_2(e_2)=(\\var{i})e_1.\\]

Write $g_2$ in matrix form.

", "correctAnswer": "matrix([0,i],[k,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

We have an $\\mathbb{R}$-linear function $h: \\mathbb{R}^2\\to \\mathbb{R}^3$ where

\\[h(e_1)=\\var{h}e_2+(\\var{i})e_3\\textrm{ and }h(e_2)=\\var{j}e_1+(\\var{k})e_2.\\]

Write $h$ in matrix form.

", "correctAnswer": "matrix([0,j],[h,k],[i,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Standard basis to matrix formv3", "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": "

Standard basis to matrix form

First decide on what size of matrix we should use.

(Hint: consider the domain and co-domain of f.)

(Hint 2: the size of the vectors that are inputs and outputs indicate the dimensions the matrix should have.)

We then want to use the FACT from Linear Algebra.

This question involves writing a function in standard basis form as an appropriately sized matrix.

We have an $\\mathbb{R}$-linear function $f: \\mathbb{R}^2\\to \\mathbb{R}^1$ where

\\[f(e_1)=\\var{d}e_1\\textrm{ and }f(e_2)=\\var{e}e_1.\\]

Write $f$ in matrix form.

We have an $\\mathbb{R}$-linear function $g_1: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_1(e_1)=\\var{f}e_1\\textrm{ and }g_1(e_2)=\\var{g}e_2.\\]

Write $g_1$ in matrix form.

We have an $\\mathbb{R}$-linear function $g_2: \\mathbb{R}^2\\to \\mathbb{R}^2$ where

\\[g_2(e_1)=\\var{k}e_1\\textrm{ and }g_2(e_2)=(\\var{i})e_1.\\]

Write $g_2$ in matrix form.

", "correctAnswer": "matrix([k,i],[0,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

We have an $\\mathbb{R}$-linear function $h: \\mathbb{R}^2\\to \\mathbb{R}^3$ where

\\[h(e_1)=\\var{h}e_1+(\\var{i})e_2\\textrm{ and }h(e_2)=\\var{j}e_1+(\\var{k})e_2.\\]

Write $h$ in matrix form.

", "correctAnswer": "matrix([h,j],[i,k],[0,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Algebraic form to matrix form", "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": "

Standard basis to matrix form

First decide on what size of matrix we should use.

(Hint: consider the domain and co-domain of f.)

(Hint 2: the size of the vectors that are inputs and outputs indicate the dimensions the matrix should have.)

We then want to use the FACT from Linear Algebra.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "l1": {"name": "l1", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "i": {"name": "i", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "l2": {"name": "l2", "group": "Ungrouped variables", "definition": "-l1", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "d2", "e", "f", "g", "h", "i", "j", "k", "l1", "l2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "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": "

This question involves writing a function given in algebraic form as an appropriately sized matrix.

Let $f$ be given by the rule

\\[f \\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}\\var{d}x+\\var{d2}y\\\\\\var{e}x\\end{pmatrix}.\\]

Write $f$ in matrix form.

", "correctAnswer": "matrix([d,d2],[e,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

Let $f$ be given by the rule

\\[f\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}\\var{f}x+\\var{g}y\\\\\\var{h}y\\end{pmatrix}.\\]

Write $f$ in matrix form.

", "correctAnswer": "matrix([f,g],[0,h])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

Let $f$ be given by the rule

\\[f\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}\\var{i}x\\\\\\var{j}x+\\var{k}y\\end{pmatrix}.\\]

Write $f$ in matrix form.

", "correctAnswer": "matrix([i,0], [j,k])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "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, "prompt": "

Let $f$ be given by the rule

\\[f\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}\\var{l1}y\\\\\\var{l2}x\\end{pmatrix}.\\]

Write $f$ in matrix form.

", "correctAnswer": "matrix([0,l1],[l2,0])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Charles Garnet's copy of Evals and evecs", "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/"}, {"name": "Charles Garnet Cox", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24585/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

This question is about finding eigenvalues of
\\[A=\\begin{pmatrix}\\var{a11}&\\var{a12}\\\\0&\\var{a22}\\end{pmatrix}.\\]

", "advice": "

A first step could be to compute the eigenvalues for the given matrix. Consider how we do this. We can then answer parts (a) and (b).

For each eigenvalue, consider the corresponding eigenvectors. Be sure to select all of the vectors that are eigenvectors of A.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(-6..-1 except ea11)", "description": "", "templateType": "anything", "can_override": false}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,a11+a22,a11-a22,a22-a11,a11,a22,ea11,ea22])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a12])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a22-a11", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..-1)", "description": "", "templateType": "anything", "can_override": false}, "ea11": {"name": "ea11", "group": "Ungrouped variables", "definition": "-a11", "description": "", "templateType": "anything", "can_override": false}, "ea22": {"name": "ea22", "group": "Ungrouped variables", "definition": "-a22", "description": "", "templateType": "anything", "can_override": false}, "ea12": {"name": "ea12", "group": "Ungrouped variables", "definition": "-a12", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a11", "a22", "a12", "a", "b", "c", "d", "ea11", "ea22", "ea12"], "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": "

The matrix $A$ has [[0]] eigenvalues.

", "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": "2", "maxValue": "2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "

The eigenvalues of $A$ are [[0]].

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Which of the following are eigenvectors of $A$? (Tick all that are.)

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Try some matrices, possibly in $M_n(\\mathbb{R})$ and $M_n(\\mathbb{C})$, to decide on each statement.

(All except for one of the answers can be deduced from examples and results from lectures.)

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Select all of the following that are always true.

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It is helpful to consider a matrix in $M_2(\\mathbb{C})$ with first column consisting only of zeroes. Can you diagonalise your matrix?

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Select all of the following that are always true for similar matrices.

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This is the third assessed quiz for MATH10015 Linear Algebra. It counts for 2.5% of the unit grade. Quizzes should be completed without discussing the questions with other people; of course it is good to use your notes, and it is fine to discuss similar problems with other students.

You can attempt this quiz as many times as you like (although questions may change on each attempt). Your score will come from your highest attempt. You should enter your answers as exact numbers unless a question instructs you to give an answer rounded to a certain number of decimal places/significant figures.

Clicking on this link will allow you to start a new attempt or resume a previous one. Please note the deadline. After you have submitted an attempt you will also see the option to review your answers to that attempt and will be able to see your mark for each question, the correct answer, and solutions for each part. Note that the questions will change for each attempt.

Once the deadline has passed you will be able to review your attempts by using the above link.

It is recommended that you take a screenshot of this page showing both your name and final score in the same screenshot in case of any technical problems.

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