Find a basis given a spanning set of vectors

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Mohamed Mehbali

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Mohamed Mehbali 7 years, 6 months ago

Created this as a copy of Find a basis given a spanning set of vectors.

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

There are 9 other versions that do you not have access to.

Question statement

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

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`

[ 1, 0 ]

← Depends on:

→ Used by:

mm
nt5
ont5

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

This question is used in the following exams:

Worksheet 1 by Mohamed Mehbali in Mohamed's workspace.
test1 by Charles Garnet Cox in Charles Garnet's workspace.

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Find a basis given a spanning set of vectors

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Taxonomy: mathcentre

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Contributors

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Checkpoint description

Mohamed Mehbali 7 years, 6 months ago

Error in variable testing condition

Error

Warning

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

History

Comment

Checkpoint description

Mohamed Mehbali 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`