Find a basis given a spanning set of vectors

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Ben Brawn

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Doesn't work

7 years, 10 months ago

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Ben Brawn 7 years, 10 months ago

Gave some feedback: Doesn't work

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Find a basis given a spanning set of vectors	Doesn't work	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Find a basis given a spanning set of vectors	draft	Mohamed Mehbali	31/08/2017 12:59
Michael's copy of Find a basis given a spanning set of vectors	draft	Michael Hoffmann	18/07/2018 23:07
John's copy of Find a basis given a spanning set of vectors	draft	John Steele	13/05/2019 03:56

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Question statement

Consider the following $6$ vectors in $\mathbb{R^4}$ . .

$\begin{align} \textbf{v}_1&=\var{rowvector(v1)}\\ \textbf{v}_2&=\var{rowvector(v2)}\\ \textbf{v}_3&=\var{rowvector(v3)}\\ \textbf{v}_4&=\var{rowvector(v4)}\\ \textbf{v}_5&=\var{rowvector(v5)}\\ \textbf{v}_6&=\var{rowvector(v6)}\end{align}$

You are given that this set of vectors is a spanning set for $\mathbb{R^4}$

Give any introductory information the student needs.

			Name	Type	Generated Value

mm5	list	[ 0, 1 ]
mm4	list	[ 1, 0 ]
nt3	string	not
mm6	list	[ 1, 0 ]
nt5	string	not
al	integer	-1
ga1	integer	-1
mm2	list	[ 0.5, 0 ]
ga	integer	1
ep	integer	0
ont3	string
ont2	string	not
ont6	string	not
ont5	string
ont4	string	not
message5	string	There are three linearly indep
message4	string
be1	integer	0
tmm	list	Nested 2×6 list
be	integer	0
nt2	string
nt4	string
v1	list	[ 4, -2, 1, 1 ]
v2	list	[ 2, -4, 5, -3 ]
v3	list	[ -4, 2, -1, -1 ]
v4	list	[ 4, -5, 4, -1 ]
mm3	list	[ 0, 1 ]
v6	list	[ -2, 1, 1, -4 ]
nt6	string
b	integer	-1
c5	integer	-3
al1	integer	-1
a	integer	-2
c	integer	-3
p1	list	[ -4, 2, -1, -1 ]
mm	list	Nested 6×2 list
q	list	[ -4, 2, -1, -1 ]
p2	list	[ -8, 7, -5, 0 ]
r	list	[ 0, -3, 3, -2 ]
u	integer	5
t	list	[ 0, 0, 0, 0 ]
v	list	[ -2, 1, 1, -4 ]
y	list	[ 2, -4, 5, -3 ]
x	list	[ 4, -2, 1, 1 ]
z	list	[ 4, -5, 4, -1 ]
v5	list	[ -8, 7, -5, 0 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(u=2 or u=5 or u=7 or u=9,[0,1], [1,0])

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ 0, 1 ]

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Match choices with answers
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Your task is to find a basis for $\mathbb{R^4}$ by finding a linearly independent subset of these vectors.

Start from $\textbf{v}_1$ and work through each vector in turn.

Determine if a vector is a linear combination of the previous vectors in the list.

If it is not such a linear combination then include it in the basis by choosing Yes, otherwise choose No.

Note that if a vector $\textbf{v}_i$ for $i=2,\ldots 5$ is a linear combination of the previous vectors in the list then it will satisfy a simple relation of the form $\textbf{v}_i=a\textbf{v}_j +b\textbf{v}_k$ where $a$ can be $0,\;1$ or $-1$ similarly for $b$ .

When you get to $\textbf{v}_6$ it will be obvious if it is in the spanning set or not. (why?)

Unnamed gap

Advice

Clearly $\textbf{v}_1$ is always in the required basis as it is non-zero.

$\textbf{v}_2$ is {nt2} in the required basis as it is {ont2} a multiple of $\textbf{v}_1$ .

$\textbf{v}_3$ is {nt3} in the required basis as it is {ont3} a linear combination of $\textbf{v}_1$ and $\textbf{v}_2$ .

$\textbf{v}_4$ is {nt4} in the required basis as it is {ont4} a linear combination of previous vectors.

{message4}

$\textbf{v}_5$ is {nt5} in the required basis as it is {ont5} a linear combination of previous vectors.

{message5}

$\textbf{v}_6$ is {nt6} in the required basis as it is {ont6} a linear combination of previous vectors.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Match choices with answers
Match choices with answers		Hasn't run yet

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Find a basis given a spanning set of vectors

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Taxonomy: mathcentre

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Ben Brawn 7 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `list`

Parts

Gap-fill

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Error

Find a basis given a spanning set of vectors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`