$W$ is non-empty
", "For all $\\boldsymbol{u} , \\boldsymbol{v} \\in W$ we have that $\\boldsymbol{u} + \\boldsymbol{v} \\in W$
", "For all $\\lambda \\in \\mathbb{R}$ and all $\\boldsymbol{u} \\in W$ we have that $\\lambda \\boldsymbol{u} \\in W$
", "For all $\\boldsymbol{u} \\in W$ we have that $- \\boldsymbol{u} \\in W$
", "For all $\\boldsymbol{u} \\in W$ we have that $\\boldsymbol{u} \\in V$
", "For all $\\boldsymbol{u} \\in W$ we have that $\\boldsymbol{u} + \\boldsymbol{u} \\in W$
"], "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "matrix": ["1", "1", "1", 0, 0, 0], "maxMarks": 0}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "Suppose we have a real vector space $V$, and let $W$ be a subset of $V$. $W$ is a subspace of $V$ if and only if it satisfies which of the three following conditions:
(Press the \"Show steps\" button if you require a hint).
[[0]]
We must find three conditions which, together, are both necessary and sufficient to ensure that W is a subspace. Only one such combination of three conditions of the above will achieve this. Any other combination of three conditions is in fact necessary for W to be a subspace, but not sufficient. Chapter 17 of the lecture notes gives the correct conditions.
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "scripts": {}, "stepsPenalty": 0, "sortAnswers": false}, {"variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "maxValue": "max([k_1 , k_2 , k_3])", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "max([k_1 , k_2 , k_3])", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}, {"maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "minAnswers": 0, "matrix": [["0", "0.5"], ["0", "0.5"], ["0.5", "0"], ["0", "0.5"]], "shuffleChoices": true, "warningType": "none", "customMarkingAlgorithm": "", "maxMarks": 0, "choices": ["$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\simplify{ x^2 + {a}*x} + \\var{b} \\sin( \\var{c} x )$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\simplify{ x^2 + {d}*x} + \\var{f} \\sin( \\var{g} x )$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\simplify{ x^2 + {h}*x} + \\var{j} \\sin( \\var{k} x )$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\simplify{ x^2 + {l}*x} + \\var{m} \\sin( \\var{n} x )$
"], "variableReplacements": [], "layout": {"expression": "", "type": "all"}, "scripts": {}, "answers": ["Yes
", "No
"], "type": "m_n_x", "displayType": "radiogroup", "shuffleAnswers": true, "marks": 0, "unitTests": [], "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}, {"maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "minAnswers": 0, "matrix": [["0", "0.5"], ["0.5", "0"], [0, "0.5"], [0, "0.5"]], "shuffleChoices": true, "warningType": "none", "customMarkingAlgorithm": "", "maxMarks": 0, "choices": ["$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\Big( \\simplify{ x^2 + {a}*x} + \\var{b} \\sin( \\var{c} x ) \\Big) + \\Big( \\simplify{ x^2 + {h}*x} + \\var{j} \\sin( \\var{k} x ) \\Big)$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\Big( \\simplify{ x^2 + {a}*x} + \\var{b} \\sin( \\var{c} x ) \\Big) +\\Big( \\simplify{ x^2 + {d}*x} + \\var{f} \\sin( \\var{g} x ) \\Big)$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\Big( \\simplify{ x^2 + {l}*x} + \\var{m} \\sin( \\var{n} x ) \\Big) +\\Big( \\simplify{ x^2 + {d}*x} + \\var{f} \\sin( \\var{g} x ) \\Big)$
", "$f: \\mathbb{R} \\longrightarrow \\mathbb{R}$ defined by $f(x) = \\Big( \\simplify{ x^2 + {l}*x} + \\var{m} \\sin( \\var{n} x ) \\Big) +\\Big( \\simplify{ x^2 + {h}*x} + \\var{j} \\sin( \\var{k} x ) \\Big)$
"], "variableReplacements": [], "layout": {"expression": "", "type": "all"}, "scripts": {}, "answers": ["Yes
", "No
"], "type": "m_n_x", "displayType": "radiogroup", "shuffleAnswers": true, "marks": 0, "unitTests": [], "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "i) Let $k \\in \\mathbb{N}$ and consider the vector space $V$ defined by $V = \\{ \\lambda_k x^k + \\lambda_{k-1} x^{k-1} + \\ldots \\lambda_1 x + \\lambda_0 \\big | \\lambda_0 , \\lambda_1 , \\ldots , \\lambda_k \\in \\mathbb{R} \\}$, with the standard operations of polynomial addition and scalar multiplication.
\nWhat is the snallest value of $k$ for which the set
$U = V \\cup \\{\\simplify{x^{{k_1}}+{c_1}} , \\simplify{x^{{k_2}}+{c_2}} , \\simplify{x^{{k_3}}+{c_3}} \\}$
is a vector space?
(Press the \"Show steps\" button if you require a hint).
$k \\geq$ [[0]] .
\nii) Now consider the vector space $V$ defined by $V = \\{ f: \\mathbb{R} \\longrightarrow \\mathbb{R} \\big| f \\text{ is differentiable and } f' (0) = 0 \\}$, with the standard operations of addition and scalar multiplication for functions.
\nNote: The easiest way to show that this is a vector space is to show that it is a subspace of the vector space consisting of all functions from $\\mathbb{R}$ to $\\mathbb{R}$, by using the subspace criterion.
\nWhich of the following are elements of $V$?
[[1]]
Which of the following are elements of $V$? Notice that each choice is a sum of two of the functions above.
[[2]]
When you have completed all gaps in this part of the question, please press the \"Show feedback\" button.
\n", "variableReplacementStrategy": "originalfirst", "steps": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "Hint for part i: We already know that $V$ is a vector space so we need only find a value of $k$ such that $V$ includes $\\simplify{x^{{k_1}}+{c_1}} , \\simplify{x^{{k_2}}+{c_2}} , \\simplify{x^{{k_3}}+{c_3}}$.
\nIf you believe you have an answer, check that all the vector space axioms hold true, and ensure that $k$ is as small as possible.
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "scripts": {"mark": {"script": "this.markingFeedback = [];\n//This clears the feedback that would normally be given automatically by the built in script. We just want to show our own feedback.\n\n//For the code below: \"this.credit tells us the proportion of the marks that the student got for this part of the question. For example, if the student scored 1 mark out of three, then \"this.credit\" would equal to 1/3.\nif (this.credit < 1){\n this.markingComment('Not quite. Try again and have a look at this feedback section once each gap has been answered correctly as there will be some useful feedback.');\n}else if (this.credit == 1){\n this.markingComment('Good! Part ii demonstrates the following fact: For a vector space $V$, we may have that some $u,v$ are not in $V$, but we may still have that $u+v$ is in $V$. More importantly, if $u+v$ is in $V$, then this does NOT necessarily mean $u$ and $v$ are in $V$.');\n}", "order": "after"}}, "stepsPenalty": 0, "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "Consider the set $V = \\{ v \\in \\mathbb{R} \\big| v > 0 \\}$.
\nWe define vector addition and scalar multiplication as follows.
For all $u,v \\in V$, vector addition is defined by numerical multiplication: $u+v := uv$
For all $\\lambda \\in \\mathbb{R}$ and all $v \\in V$, scalar multiplication is defined by numerical exponentiation: $\\lambda v := v^\\lambda$.
It can be shown that the set $V$, with these operations of vector addition and scalar multiplication, satisfies the vector space axioms.
i) What element of $V$ plays the role of the zero vector? [[0]]
\nii) For a given $v \\in V$, what element of $V$ plays the role of \"negative $v$\"? [[1]]
", "gaps": [{"correctAnswerFraction": false, "maxValue": "1", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReduced": false, "showCorrectAnswer": false, "minValue": "1", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}, {"variableReplacements": [], "vsetRangePoints": 5, "checkingType": "absdiff", "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "marks": 1, "unitTests": [], "customMarkingAlgorithm": "", "answer": "1/v", "vsetRange": [0, 1], "showPreview": true, "failureRate": 1, "showCorrectAnswer": false, "checkVariableNames": false, "type": "jme", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}], "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "Let $V$ be a real vector space, and let $\\{ \\boldsymbol{v_1} , \\boldsymbol{v_2} , \\ldots , \\boldsymbol{v_k} \\}$ be a set of $k$ vectors in $V$.
The set $\\{ \\boldsymbol{v_1} , \\boldsymbol{v_2} , \\ldots , \\boldsymbol{v_k} \\}$ is linearly independent if and only which one of the following conditions holds?
[[0]]
The set $\\{\\boldsymbol{v_1} ,\\boldsymbol{v_2} , \\ldots ,\\boldsymbol{v_k} \\}$ spans $V$ if and only if which one of the following conditions holds?
[[1]]
The only solution to $\\lambda_1 \\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k} = 0$ is the trivial solution: $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k = 0$.
", "At least one of the solutions to $\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k} = 0$ is the trivial solution: $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k = 0$.
", "There is no solution to $\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k} = 0$.
", "For all $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k \\in \\mathbb{R}$, we have that $\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k} = 0$.
"], "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "maxMarks": 0}, {"distractors": ["", "", "", ""], "type": "1_n_2", "unitTests": [], "displayColumns": "1", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "displayType": "radiogroup", "customMarkingAlgorithm": "", "variableReplacements": [], "minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["Every vector $\\boldsymbol{v} \\in V$ can be written as a linear combination $\\boldsymbol{v} =\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k}$, for some $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k \\in \\mathbb{R}$.
", "Every vector $\\boldsymbol{v} \\in V$ can be written as a linear combination $\\boldsymbol{v} =\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k}$, for some $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k \\in \\mathbb{R}$ which are all distinct.
", "Every vector $\\boldsymbol{v} \\in V$ can be written as a linear combination $\\boldsymbol{v} =\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k}$, for some $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k \\in \\mathbb{R}$ which are all non-zero.
", "Every vector $\\boldsymbol{v} \\in V$ can be written as a linear combination $\\boldsymbol{v} =\\lambda_1\\boldsymbol{v_1} + \\lambda_2 \\boldsymbol{v_2} + \\ldots + \\lambda_k \\boldsymbol{v_k}$ for some $\\lambda_1 , \\lambda_2 , \\ldots , \\lambda_k \\in \\mathbb{R}$ for which at least one is non-zero.
"], "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "maxMarks": 0}], "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "For each one of the following sets, determine if it 1) only spans, 2) is only linearly independent, 3) both spans and is linearly independent, 4) neither spans nor is linearly independent.
\n[[0]]
", "gaps": [{"maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "minAnswers": 0, "matrix": [["0.5", 0, 0, 0], [0, "0.5", 0, 0], [0, 0, "0.5", 0], [0, 0, 0, "0.5"], [0, 0, 0, "0.5"], [0, 0, 0, "0.5"]], "shuffleChoices": true, "warningType": "none", "customMarkingAlgorithm": "", "maxMarks": 0, "choices": ["$\\Bigg\\{ \\var{{V3_1}} , \\var{{V3_2}} , \\var{{V3_3}} \\Bigg\\}$
", "$\\Bigg\\{ \\var{{V3_1}} , \\var{{V3_1pV3_2}} , \\var{{V3_1pV3_2pV3_3}} , \\var{{V3_1pV3_3}} \\Bigg\\}$
", "$\\Bigg\\{ \\var{{V3_1pV3_2pV3_3}} \\Bigg\\}$
", "$\\Bigg\\{ \\var{{V3_2pV3_3}} , \\var{{V3_2t2pV3_3t2}} \\Bigg\\}$
", "$\\Bigg\\{ \\var{{V3_1pV3_3}} , \\var{{V3_1mV3_3t2}} , \\var{{V3_1t2mV3_3}} \\Bigg\\}$
", "$\\Bigg\\{ \\var{{V3_1pV3_2}} , \\var{{V3_1mV3_2}} , \\var{{V3_1mV3_2t2}} , \\var{{V3_1t2mV3_2}} \\Bigg\\}$
"], "variableReplacements": [], "layout": {"expression": "", "type": "all"}, "scripts": {}, "answers": ["Both spans and is linearly independent
", "only spans
", "linearly independent only
", "neither
"], "type": "m_n_x", "displayType": "radiogroup", "shuffleAnswers": false, "marks": 0, "unitTests": [], "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "sortAnswers": false}, {"variableReplacements": [], "gaps": [{"variableReplacements": [], "vsetRangePoints": 5, "checkingType": "absdiff", "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "marks": "5", "unitTests": [], "customMarkingAlgorithm": "", "answer": "({c1}/{a1})*a + (-({c1}*{d1})/({a1}*{e1}) + {f1}/{e1}) b", "vsetRange": [0, 1], "showPreview": true, "failureRate": 1, "showCorrectAnswer": true, "checkVariableNames": false, "type": "jme", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "marks": 0, "unitTests": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "Let $\\boldsymbol{v} = \\begin{pmatrix} a \\\\ b \\\\ c \\end{pmatrix} \\in \\mathbb{R}$. Find an equation that $a , b , c$ must satisfy so that $\\boldsymbol{v}$ can be expressed as a linear combination of the vectors $\\var{{V1}}$ and $\\var{{V2}}$. (If you wish, you may press the \"Show steps\" button to answer this question in steps).
\n$c =$ [[0]] .
", "variableReplacementStrategy": "originalfirst", "steps": [{"correctAnswer": "matrix([a1 , d1] , [0 , e1] , [c1 , f1])", "allowResize": false, "type": "matrix", "variableReplacementStrategy": "originalfirst", "correctAnswerFractions": false, "tolerance": 0, "marks": 1, "allowFractions": false, "unitTests": [], "numRows": "3", "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "In order for $\\boldsymbol{v}$ to be expressable as a linear combination of the vectors $\\var{{V1}}$ and $\\var{{V2}}$, there must exist $x,y \\in \\mathbb{R}$ such that
$\\begin{pmatrix} a \\\\ b \\\\ c \\end{pmatrix} = x \\var{{V1}} + y \\var{{V2}}$. This can be expressed in matrix form as $A \\boldsymbol{x} = \\boldsymbol{v}$, where $\\boldsymbol{x} = \\begin{pmatrix} x \\\\ y \\end{pmatrix}$ and $A=$
For such $x,y$ to exist, we need this system to be consistent. We should find the reduced row echelon form of $(A | \\boldsymbol{v})$ to help us determine when the system is consistent.
\nConsider the row operations $R_1 \\rightarrow \\simplify{1/{a1}}R_1$, $R_3 \\rightarrow \\simplify{R_3 - {c1} R_1}$, $R_2 \\rightarrow \\simplify{1/{e1}}R_2$, $R_3 \\rightarrow \\simplify{R_3 - ({f1}-{c1}*{d1}/{a1}) R_2}$, $R_1 \\rightarrow \\simplify{R_1 - {d1}/{a1} R_2}$.
In an attempt to find the reduced row echelon form of $(A | \\boldsymbol{v})$, we can perform these row operations (in the given order) on $(A | \\boldsymbol{v}) = \\begin{pmatrix} \\var{{a1}} & \\var{{d1}} & a \\\\ 0 &\\var{{e1}} & b \\\\ \\var{{c1}} &\\var{{f1}} & c \\end{pmatrix}$ and we obtain the matrix $\\textbf{RRE} (A | \\boldsymbol{v}) = \\begin{pmatrix} 1 & 0 & f_1 (a,b,c) \\\\ 0 & 1 & f_2 (a,b,c) \\\\ 0 & 0 & f_3 (a,b,c) \\end{pmatrix}$, where $f_1 (a,b,c) , f_2 (a,b,c) , f_3 (a,b,c)$ are some expressions of $a,b,c$. What are these expressions explicitely? (If you wish to write something like $2a$ or $ab$ then please write 2*a or a*b and not 2a or ab, respectively, as the system may not correctly interpret the latter two).
$f_1 (a,b,c) =$
"}, {"answerSimplification": "all", "vsetRangePoints": 5, "checkingType": "absdiff", "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "marks": 1, "unitTests": [], "customMarkingAlgorithm": "", "answer": "(1/{e1})*b", "vsetRange": [0, 1], "showPreview": true, "failureRate": 1, "variableReplacements": [], "showCorrectAnswer": true, "checkVariableNames": false, "type": "jme", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "prompt": "$f_2 (a,b,c) =$
"}, {"answerSimplification": "all", "vsetRangePoints": 5, "checkingType": "absdiff", "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "marks": 1, "unitTests": [], "customMarkingAlgorithm": "", "answer": "(-{c1}/{a1})*a + (({c1}*{d1})/({a1}*{e1}) - {f1}/{e1}) b + c", "vsetRange": [0, 1], "showPreview": true, "failureRate": 1, "variableReplacements": [], "showCorrectAnswer": true, "checkVariableNames": false, "type": "jme", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "prompt": "$f_3 (a,b,c) =$
"}, {"answerSimplification": "all", "vsetRangePoints": 5, "checkingType": "absdiff", "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "marks": 1, "unitTests": [], "customMarkingAlgorithm": "", "answer": "({c1}/{a1})*a + (-({c1}*{d1})/({a1}*{e1}) + {f1}/{e1}) b", "vsetRange": [0, 1], "showPreview": true, "failureRate": 1, "variableReplacements": [], "showCorrectAnswer": true, "checkVariableNames": false, "type": "jme", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "prompt": "Recall that in order for this system to be consistent, we require that the row rank of $A$ be equal to the row rank of $(A|\\boldsymbol{b})$. For this to be the case here, we must have
$c=$
Week 9 (lectures 17 and 18): In this question, you will look at vector spaces, subspaces, linearly independent sets, and spanning sets.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
See the question description for an explanation of the variables of part b.
", "definition": "random(2..4)", "group": "Variables for part b"}, "v": {"name": "v", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "-u", "group": "Variables for part b"}, "l": {"name": "l", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "r-m*n", "group": "Variables for part b"}, "V3_1t2mV3_3": {"name": "V3_1t2mV3_3", "templateType": "anything", "description": "", "definition": "2*V3_1 - V3_3", "group": "variables for part e"}, "V3_1pV3_3": {"name": "V3_1pV3_3", "templateType": "anything", "description": "", "definition": "V3_1 + V3_3", "group": "variables for part e"}, "k_1": {"name": "k_1", "templateType": "anything", "description": "", "definition": "random(k_0 -3..k_0)", "group": "Variables for part b"}, "a2": {"name": "a2", "templateType": "anything", "description": "This will form part of one of our matrices in part e.
", "definition": "random(-5..5 except(0))", "group": "variables for part e"}, "k": {"name": "k", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-4..4 except(0) except(-1) except(1))", "group": "Variables for part b"}, "a1": {"name": "a1", "templateType": "anything", "description": "These will form part of our vectors in part f.
", "definition": "random(-5..5 except(0))", "group": "variables for part f"}, "k_0": {"name": "k_0", "templateType": "anything", "description": "", "definition": "random(4..20)", "group": "Variables for part b"}, "f1": {"name": "f1", "templateType": "anything", "description": "These will form part of our vectors in part f.
", "definition": "random(-5..5 except(0))", "group": "variables for part f"}, "c_1": {"name": "c_1", "templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Variables for part b"}, "d1": {"name": "d1", "templateType": "anything", "description": "These will form part of our vectors in part f.
", "definition": "random(-5..5 except(0))", "group": "variables for part f"}, "c2": {"name": "c2", "templateType": "anything", "description": "This will form part of one of our matrices in part e.
", "definition": "random(-5..5 except(0))", "group": "variables for part e"}, "c": {"name": "c", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-4..4 except(0) except(-1) except(1))", "group": "Variables for part b"}, "f": {"name": "f", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(2..4)", "group": "Variables for part b"}, "V3_1mV3_2": {"name": "V3_1mV3_2", "templateType": "anything", "description": "", "definition": "V3_1 - V3_2", "group": "variables for part e"}, "V3_1mV3_3t2": {"name": "V3_1mV3_3t2", "templateType": "anything", "description": "", "definition": "V3_1 - 2*V3_3", "group": "variables for part e"}, "V3_2": {"name": "V3_2", "templateType": "anything", "description": "This is one of our vectors for part e. Notice that it is linearly independent to V3_1.
", "definition": "vector(a2 , b2 + e2 , c2 + f2)", "group": "variables for part e"}, "f2": {"name": "f2", "templateType": "anything", "description": "This will form part of one of our matrices in part e.
", "definition": "random(-5..5 except(0))", "group": "variables for part e"}, "d": {"name": "d", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "v-f*g", "group": "Variables for part b"}, "V2": {"name": "V2", "templateType": "anything", "description": "This is the second vector of part f.
", "definition": "matrix([d1] , [e1] , [f1])", "group": "variables for part f"}, "V3_1t2mV3_2": {"name": "V3_1t2mV3_2", "templateType": "anything", "description": "", "definition": "2*V3_1 - V3_2", "group": "variables for part e"}, "b2": {"name": "b2", "templateType": "anything", "description": "This will form part of one of our matrices in part e.
", "definition": "random(-5..5 except(0))", "group": "variables for part e"}, "V3_1pV3_2": {"name": "V3_1pV3_2", "templateType": "anything", "description": "", "definition": "V3_1 + V3_2", "group": "variables for part e"}, "V1": {"name": "V1", "templateType": "anything", "description": "This is the first vector of part f.
", "definition": "matrix([a1] , [0] , [c1])", "group": "variables for part f"}, "V3_1": {"name": "V3_1", "templateType": "anything", "description": "This is one of our matrices in part e.
", "definition": "vector(a2 , b2 , c2)", "group": "variables for part e"}, "e1": {"name": "e1", "templateType": "anything", "description": "These will form part of our vectors in part f.
", "definition": "random(-5..5 except(0))", "group": "variables for part f"}, "b": {"name": "b", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(2..4)", "group": "Variables for part b"}, "n": {"name": "n", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-4..4 except(0) except(-1) except(1))", "group": "Variables for part b"}, "c1": {"name": "c1", "templateType": "anything", "description": "These will form part of our vectors in part f.
", "definition": "random(-5..5 except(0))", "group": "variables for part f"}, "w": {"name": "w", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "0", "group": "Variables for part b"}, "V3_2t2pV3_3t2": {"name": "V3_2t2pV3_3t2", "templateType": "anything", "description": "", "definition": "2*V3_2 + 2*V3_3", "group": "variables for part e"}, "k_3": {"name": "k_3", "templateType": "anything", "description": "", "definition": "random(k_0 -3..k_0)", "group": "Variables for part b"}, "a": {"name": "a", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "u-b*c", "group": "Variables for part b"}, "c_2": {"name": "c_2", "templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Variables for part b"}, "u": {"name": "u", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-5..5 except(0))", "group": "Variables for part b"}, "c_3": {"name": "c_3", "templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Variables for part b"}, "m": {"name": "m", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(2..4)", "group": "Variables for part b"}, "g": {"name": "g", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-4..4 except(0) except(-1) except(1))", "group": "Variables for part b"}, "V3_3": {"name": "V3_3", "templateType": "anything", "description": "This is the cross product of V3_1 and V3_2. This was done so that V3_1 , V3_2 , V3_2 are linearly independent. To see how our vectors are created for part e, see the description of the question.
", "definition": "cross(V3_1 , V3_2)", "group": "variables for part e"}, "e2": {"name": "e2", "templateType": "anything", "description": "This will form part of one of our matrices in part e.
", "definition": "random(-5..5 except(0))", "group": "variables for part e"}, "r": {"name": "r", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "random(-5..5 except(0) except(u) except(-u))", "group": "Variables for part b"}, "V3_2pV3_3": {"name": "V3_2pV3_3", "templateType": "anything", "description": "", "definition": "V3_2 + V3_3", "group": "variables for part e"}, "h": {"name": "h", "templateType": "anything", "description": "See the question description for an explanation of the variables of part b.
", "definition": "w-j*k", "group": "Variables for part b"}, "k_2": {"name": "k_2", "templateType": "anything", "description": "", "definition": "random(k_0 -3..k_0)", "group": "Variables for part b"}, "V3_1mV3_2t2": {"name": "V3_1mV3_2t2", "templateType": "anything", "description": "", "definition": "V3_1 - 2*V3_2", "group": "variables for part e"}}, "metadata": {"description": "This is the question for week 9 of the MA100 course at the LSE. It looks at material from chapters 17 and 18.
Description of variables for part b:
For part b we want to have four functions such that the derivative of one of them, evaluated at 0, gives 0; but for the rest we do not get 0. We also want two of the ones that do not give 0, to be such that the derivative of their sum, evaluated at 0, gives 0; but when we do this for any other sum of two of our functions, we do not get 0. Ultimately this part of the question will show that even if two functions are not in a vector space (the space of functions with derivate equal to 0 when evaluated at 0), then their sum could nonetheless be in that vector space. We want variables which statisfy:
a,b,c,d,f,g,h,j,k,l,m,n are variables satisfying
Function 1: x^2 + ax + b sin(cx)
Function 2: x^2 + dx + f sin(gx)
Function 3: x^2 + hx + j sin(kx)
Function 4: x^2 + lx + m sin(nx)
u,v,w,r are variables satifying
u=a+bc
v=d+fg
w=h+jk
r=l+mn
The derivatives of each function, evaluated at zero, are:
Function 1: u
Function 2: v
Function 3: w
Function 4: r
So we will define
u as random(-5..5 except(0))
v as -u
w as 0
r as random(-5..5 except(0) except(u) except(-u))
Then the derivative of function 3, evaluated at 0, gives 0. The other functions give non-zero.
Also, the derivative of function 1 + function 2 gives 0. The other combinations of two functions give nonzero.
We now take b,c,f,g,j,k,m,n to be defined as \\random(-3..3 except(0)).
We then define a,d,h,l to satisfy
u=a+bc
v=d+fg
w=h+jk
r=l+mn
Description for variables of part e:
Please look at the description of each variable for part e in the variables section, first.
As described, the vectors V3_1 , V3_2 , V3_3 are linearly independent. We will simply write v1 , v2 , v3 here.
In part e we ask the student to determine which of the following sets span, are linearly independent, are both, are neither:
both: v1,v2,v3
span: v1,v1+v2,v1+v2+v3, v1+v2+v3,2*v1+v2+v3
lin ind: v1+v2+v3
neither: v2+v3 , 2*v2 + 2*v3
neither:v1+v3,v1-2*v3,2*v1-v3
neither: v1+v2,v1-v2,v1-2*v2,2*v1-v2