// Numbas version: finer_feedback_settings {"name": "Week 10 (non-assessed)", "metadata": {"description": "", "licence": "All rights reserved"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "true false inner products", "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": "", "advice": "

Do check the lecture notes for these statements.

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Select all of the following that are always true.

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Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}1&0&1\\\\0&0&i\\\\1&-i&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}i&0&1\\\\0&3&i\\\\1&-i&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}5&2&1+i\\\\2&1&0\\\\1-i&0&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}i&0&0\\\\0&-i&0\\\\0&0&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}i&0\\\\0&i\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes. There is a problem sheet question that may also be helpful.

Let $A=\\begin{pmatrix}0&2e^i\\\\2e^{-i}&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes. There is a problem sheet question that may also be helpful.

Let $A=\\begin{pmatrix}0&2e^i\\\\2&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes. There is a problem sheet question that may also be helpful.

Let $A=\\begin{pmatrix}0&-2e^i\\\\2e^{i}&0\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes.

Let $A=\\begin{pmatrix}e^{i}&0\\\\0&e^{-i}\\end{pmatrix}$. Select all of the following that are true.

Check the definitions and examples within the lecture notes. There is a problem sheet question that may also be helpful.

Let $A=\\begin{pmatrix}e^i&0\\\\0&3e^{-i}\\end{pmatrix}$. Select all of the following that are true.

There is a tutorial activity this week where these different properties will be compared, but producing examples with each property will help understand how they are related.

Select all of the following that are true.

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Let $V=\\mathbb{C}^2$ over $\\mathbb{C}$ be equipped with the standard complex inner product. Let $v_1=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}1\\\\1\\end{pmatrix}$ and $v_2=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}1\\\\-1\\end{pmatrix}$.

", "advice": "

Do check the definition of basis and orthonormal basis. Note that we are using the standard complex inner product.

We can then find the required matrix as seen in lectures.

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The set $A=\\{v_1, v_2\\}$ is a basis. [[0]]

The set $A=\\{v_1, v_2\\}$ is an orthonormal basis. [[1]]

Let $f: V\\to V$ be a linear map. With $M_{AA}(f):=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}$, fill in the correct formulae for the following.

$a=\\langle$[[0]], [[1]]$\\rangle$

$b=\\langle$[[2]], [[3]]$\\rangle$

$c=\\langle$[[4]], [[5]]$\\rangle$

$d=\\langle$[[6]], [[7]]$\\rangle$

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Define $f:V\\to V$ by $f\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}x\\\\x+y\\end{pmatrix}$.

Hence $M_{AA}(f)=$[[0]].

Note that the answer can be checked by seeing whether $f(v_1)$ is $\\begin{pmatrix}a\\\\c\\end{pmatrix}_A$ where $\\begin{pmatrix}a\\\\c\\end{pmatrix}$ is the first column of $M_{AA}(f)$.

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", "advice": "

Do check the definition of basis and orthonormal basis. Note that we are using the standard complex inner product.

We can then find the required matrix as seen in lectures.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "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 set $A=\\{v_1, v_2\\}$ is a basis. [[0]]

The set $A=\\{v_1, v_2\\}$ is an orthonormal basis. [[1]]

Let $f: V\\to V$ be a linear map. With $M_{AA}(f):=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}$, fill in the correct formulae for the following.

$a=\\langle$[[0]], [[1]]$\\rangle$

$b=\\langle$[[2]], [[3]]$\\rangle$

$c=\\langle$[[4]], [[5]]$\\rangle$

$d=\\langle$[[6]], [[7]]$\\rangle$

Define $f:V\\to V$ by $f\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}x+y\\\\x-y\\end{pmatrix}$.

Hence $M_{AA}(f)=$[[0]].

Note that the answer can be checked by seeing whether $f(v_1)$ is $\\begin{pmatrix}a\\\\c\\end{pmatrix}_A$ where $\\begin{pmatrix}a\\\\c\\end{pmatrix}$ is the first column of $M_{AA}(f)$.

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

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.

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