// Numbas version: finer_feedback_settings {"name": "Week 12 (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 rank nullity", "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": "

Write out the formula that the Rank Nullity Theorem gives us. How can we use this in relation to these questions (the equivalent characterisations for injective and surjective for linear maps are important here).

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Select all of the following that are always true.

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Consider how, in lectures, we came to understand when vector spaces are isomorphic (and when they are not).

Select all of the following that are always true.

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Let $\\Psi: \\mathbb{P}\\to \\mathbb{R}$ be defined by sending each element of $\\{1, x, x^2, \\ldots\\}$ to 1 and assume that $\\Psi$ is $\\mathbb{R}$-linear.

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Consider the image of various points under $\\Psi$. Of importance here is that $\\Psi$ is $\\mathbb{R}$-linear.

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For $p(x)=\\var{a3}$, we have that $\\Psi(p)=$[[3]].

For $p(x)=\\var{a2}+\\var{c2}x^2$, we have that $\\Psi(p)=$[[2]].

For $p(x)=\\var{a1}+\\var{b1}x+\\var{c1}x^2$, we have that $\\Psi(p)=$[[1]].

For $p(x)=\\var{a}+\\var{b}x+\\var{c}x^2+\\var{d}x^3$, we have that $\\Psi(p)=$[[0]].

Summarise, in your own words, how to find $\\Psi(p)$ for a general $p\\in \\mathbb{P}$.

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Let $\\Psi: \\mathbb{P}\\to \\mathbb{R}$ be defined by sending each element of $\\{1, x, x^2, \\ldots\\}$ to 1 and assume that $\\Psi$ is $\\mathbb{R}$-linear.

Define $S\\subseteq \\mathbb{P}$ to consist of all polynomials with coefficients from $\\{0, 1\\}$.

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Again, the definition of $\\Psi$ is of great importance here. Understanding what it does to general elements in $\\mathbb{P}$ will help.

Select all of the following that are always true.

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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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