Do three given vectors form a spanning set and are they linearly independent?

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Do three given vectors form a spanning set and are they linearly independent?

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Do three given vectors form a spanning set and are they linearly independent?

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Generated value: vector

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Newcastle University Mathematics and Statistics

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Name	Status	Author	Last Modified
Do three given vectors form a spanning set and are they linearly independent?	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Simon's copy of Do three given vectors form a spanning set and are they linearly independent?	draft	Simon Thomas	12/06/2019 15:09

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Question statement

You are given the following three vectors in $\mathbb{R}^4$ :

$\mathbf{v}_1=\var{v1}, \quad \mathbf{v}_2=\var{v2}, \quad \mathbf{v}_3=\var{v3}$

Give any introductory information the student needs.

			Name	Type	Generated Value

v1	vector	vector(13,-5,-10,15)
v2	vector	vector(5,-13,6,-1)
v3	vector	vector(-18,18,4,-14)
tt	list	[ 3, 1, 2 ]

			Name	Type	Generated Value

cannot	string
only	string	amongst an infinite number of
independent	string	dependent

			Name	Type	Generated Value

is_independent	boolean	false
t	integer	3
rs	list	[ 8, 7, 2 ]
r1	number	-8
r2	number	7
r3	number	2
w_x	integer	-1
w_y	integer	-1

			Name	Type	Generated Value

marking_matrix_independent

list

[ 0, 1 ]

			Name	Type	Generated Value

w	vector	vector(-18,18,4,-14)
x	vector	vector(13,-5,-10,15)
y	vector	vector(5,-13,6,-1)
z	vector	vector(-9,-6,13,-15)

			Name	Type	Generated Value

p	integer	-1
q	integer	-1
coeff_v1	integer	1
coeff_v2	integer	1
coeff_v3	integer	1

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
switch(t=1,w, x)

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?

vector(13,-5,-10,15)

← Depends on:

→ Used by:

test

This variable doesn't seem to be used anywhere.

Part a) Gap-fill
1. Gap Gap 0. Choose one from a list
  Add
2. Gap Gap 1. Choose one from a list
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

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

1. Do these vectors form a spanning set for $\mathbb{R^4}$ ? Gap 0.

2. Is $\{\mathbf{v}_1,\;\mathbf{v}_2,\;\mathbf{v}_3\}$ a linearly independent set of vectors?Gap 1.

Advice

1. These vectors cannot form a spanning set as you need at least 4 vectors to form a spanning set in $\mathbb{R^4}$ .

2. $\mathbf{v}_1, \; \mathbf{v}_2 , \; \mathbf{v}_3$ is a linearly independent set if the only solution for

$a \mathbf{v}_1 + b \mathbf{v}_2 + c \mathbf{v}_3 = \var{vector(0,0,0,0)} \qquad \textbf{(1)}$ is

$a=0$ ,

$b=0$ ,

$c=0$ .

If there is any other solution then the set is linearly dependent. Note that if there is one other solution then there will be an infinite number of such solutions.

On writing out the components given by equation (1), we see that we get the following four equations:

$\begin{align} \simplify[std]{{v1[0]} * a + {v2[0]} * b + {v3[0]} * c }&= 0\\ \simplify[std]{{v1[1]} * a + {v2[1]} * b + {v3[1]} * c }&= 0\\ \simplify[std]{{v1[2]} *a+ {v2[2]} * b + {v3[2]} * c }&= 0\\ \simplify[std]{{v1[3]} *a+ {v2[3]} * b + {v3[3]} * c }&= 0\end{align}$

We see that on solving these equations that there is {only} the solution:

$a = \var{coeff_v1}, \; b=\var{coeff_v2}, \; c=\var{coeff_v3}$

It follows that $\{\mathbf{v}_1, \; \mathbf{v}_2 , \; \mathbf{v}_3\}$ is a linearly {independent} set of vectors.

Because the vectors $\mathbf{v}_1$ , $\mathbf{v}_2$ and $\mathbf{v}_3$ are linearly independent, input 0 for both $p$ and $q$ .Rearrange equation (1) to give

$\var{[a,b,c][tt[0]-1]}\mathbf{v}_{\var{tt[0]}} = -\var{[a,b,c][tt[1]-1]}\mathbf{v}_{\var{tt[1]}} - \var{[a,b,c][tt[2]-1]}\mathbf{v}_{\var{tt[2]}}$ So, using the solution found above,

$p=-\var{[a,b,c][tt[1]-1]} = \var{p}$ and

$q =-\var{[a,b,c][tt[2]-1]} = \var{q}$ .

Give a worked solution to the whole question.

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This question is used in the following exams:

vector(0,0,0,0)