Simon's copy of Determine if vectors form a spanning set

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Simon Thomas 5 years, 10 months ago

Created this as a copy of Determine if vectors form a spanning set.

Name	Status	Author	Last Modified
Determine if vectors form a spanning set	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Simon's copy of Determine if vectors form a spanning set	draft	Simon Thomas	12/06/2019 15:02
Ugur's copy of Determine if vectors form a spanning set	Ready to use	Ugur Efem	13/10/2024 20:15

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Question statement

Consider the following $5$ vectors in $\mathbb{R^4}$ .

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

Give any introductory information the student needs.

			Name	Type	Generated Value

f1	integer	-1
f2	integer	0
f3	integer	0
f4	integer	0
another	string	There is one other simple rela
eg	string	This is one of the relations.
al	integer	-1
ga1	integer	0
are	string	are
ga	integer	0
ep	integer	0
es	string	s
contains	string	does not contain
be1	integer	1
nt	string	not
be	integer	1
v1	list	[ -1, 2, -4, 2 ]
v2	list	[ 1, -2, 4, -2 ]
v3	list	[ -2, 1, -2, 0 ]
v4	list	[ -4, 2, -1, -2 ]
v5	list	[ -3, 3, -6, 2 ]
u	integer	2
b	integer	1
c5	integer	3
al1	integer	1
a	integer	-1
c	integer	3
p1	list	[ 1, -2, 4, -2 ]
thismany	integer	2
mm	list	[ 0, 1 ]
t0	integer	2
q	list	[ -1, -1, 2, -2 ]
p2	list	[ -3, 3, -6, 2 ]
r	list	[ -1, -1, 2, -2 ]
t	list	[ 0, 0, 0, 0 ]
v	list	[ 2, -4, 2, 1 ]
y	list	[ -2, 1, -2, 0 ]
x	list	[ -1, 2, -4, 2 ]
z	list	[ -4, 2, -1, -2 ]
test	string	$v_2+v_5=v_3$

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(u<9,al,0)

JME function reference

Description

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

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Generated value: `integer`

-1

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill

Gap-fill

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

1. Is $\{\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4,\;\textbf{v}_5\}$ a linearly independent set of vectors? Gap 0.

2. Do the above vectors form a spanning set of $\mathbb{R}^4$ ? Gap 1.

3. Does the set $\{\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4,\;\textbf{v}_5\}$ contain a linearly independent subset which forms a basis of $\mathbb{R}^4$ ? Gap 2.

Advice

(a)

Not linearly independent as any set of more than $4$ vectors in $\mathbb{R^4}$ is linearly dependent.

(b)

Note that

${v}_{\var{t0}} =\simplify{ {f1} * v_1+ {f2} * v_2 + {f3} * v_3 + {f4} * v_4}.$

Note also {test}

Because 2 of our 5 vectors can be written as linear combinations of the other 3, there are not 4 linearly independent vectors in the list. Hence not all vectors in $\mathbb{R^4}$ can be written as a linear combination of these vectors, so they are not spanning.

(c)

This set {contains} a linearly independent subset of $4$ vectors as it is {nt} spanning.

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This question is used in the following exam:

Linear algebra by Simon Thomas in Maths support.

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Simon's copy of Determine if vectors form a spanning set

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Taxonomy: mathcentre

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Simon Thomas 5 years, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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Simon's copy of Determine if vectors form a spanning set

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`