Prove vector sets are linearly independent

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If $X_1,X_2$ and $X_3$ are vectors in $mathbbR^3$ such that $X_1,X_2$ and $X_1,X_3$ are linearly independent sets, then $X_1,X_2,X_3$ is a linearly independent set or not?

since linearly independent sets need to have consistent solution and no variable can be zero. so i am assuming that in $X_1,X_2,X_3$ , $X_2$ could make all variables zero making it linearly dependent? i am not sure it i am correct.

linear-algebra vectors

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edited Jul 23 at 17:30

mrtaurho

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asked Jul 23 at 17:18

Ouila SaRaH

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up vote
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If $X_1,X_2$ and $X_3$ are vectors in $mathbbR^3$ such that $X_1,X_2$ and $X_1,X_3$ are linearly independent sets, then $X_1,X_2,X_3$ is a linearly independent set or not?

since linearly independent sets need to have consistent solution and no variable can be zero. so i am assuming that in $X_1,X_2,X_3$ , $X_2$ could make all variables zero making it linearly dependent? i am not sure it i am correct.

linear-algebra vectors

share|cite|improve this question

edited Jul 23 at 17:30

mrtaurho

750219

asked Jul 23 at 17:18

Ouila SaRaH

31

add a comment | 

up vote
-1
down vote

favorite

up vote
-1
down vote

favorite

If $X_1,X_2$ and $X_3$ are vectors in $mathbbR^3$ such that $X_1,X_2$ and $X_1,X_3$ are linearly independent sets, then $X_1,X_2,X_3$ is a linearly independent set or not?

since linearly independent sets need to have consistent solution and no variable can be zero. so i am assuming that in $X_1,X_2,X_3$ , $X_2$ could make all variables zero making it linearly dependent? i am not sure it i am correct.

linear-algebra vectors

share|cite|improve this question

edited Jul 23 at 17:30

mrtaurho

750219

asked Jul 23 at 17:18

Ouila SaRaH

31

If $X_1,X_2$ and $X_3$ are vectors in $mathbbR^3$ such that $X_1,X_2$ and $X_1,X_3$ are linearly independent sets, then $X_1,X_2,X_3$ is a linearly independent set or not?

since linearly independent sets need to have consistent solution and no variable can be zero. so i am assuming that in $X_1,X_2,X_3$ , $X_2$ could make all variables zero making it linearly dependent? i am not sure it i am correct.

linear-algebra vectors

share|cite|improve this question

edited Jul 23 at 17:30

mrtaurho

750219

asked Jul 23 at 17:18

Ouila SaRaH

31

share|cite|improve this question

share|cite|improve this question

edited Jul 23 at 17:30

mrtaurho

750219

edited Jul 23 at 17:30

mrtaurho

750219

edited Jul 23 at 17:30

mrtaurho

750219

750219

asked Jul 23 at 17:18

Ouila SaRaH

31

asked Jul 23 at 17:18

Ouila SaRaH

31

asked Jul 23 at 17:18

Ouila SaRaH

31

31

add a comment | 

add a comment | 

3 Answers
3

active

oldest

votes

up vote
0
down vote



accepted

Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_3=2X_2$ for example.

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answered Jul 23 at 17:29

Foobaz John

18k41245

add a comment | 

up vote
0
down vote

Guide:

• Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.




• What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

• It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

add a comment | 

up vote
0
down vote

Only because the two sets of $X_1,X_2$ and $X_1,X_3$ does not guarantees you that either $X_1,X_2,X_3$ or $X_2,X_3$ are linear indenpent. Just consider the three vectors

$$X_1~=~beginpmatrix1\2\3endpmatrix X_2~=~beginpmatrix4\5\6endpmatrix X_3~=~beginpmatrix8\10\12endpmatrix$$

As you can see you restrictions are fullfilled but neither the set $X_2,X_3$ nor the whole set $X_1,X_2,X_3$ are linear independent.

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

add a comment | 

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3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_3=2X_2$ for example.

share|cite|improve this answer

answered Jul 23 at 17:29

Foobaz John

18k41245

add a comment | 

up vote
0
down vote



accepted

Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_3=2X_2$ for example.

share|cite|improve this answer

answered Jul 23 at 17:29

Foobaz John

18k41245

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_3=2X_2$ for example.

share|cite|improve this answer

answered Jul 23 at 17:29

Foobaz John

18k41245

Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_3=2X_2$ for example.

share|cite|improve this answer

answered Jul 23 at 17:29

Foobaz John

18k41245

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 17:29

Foobaz John

18k41245

answered Jul 23 at 17:29

Foobaz John

18k41245

answered Jul 23 at 17:29

Foobaz John

18k41245

18k41245

add a comment | 

add a comment | 

up vote
0
down vote

Guide:

• Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.




• What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

• It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

add a comment | 

up vote
0
down vote

Guide:

• Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.




• What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

• It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

add a comment | 

up vote
0
down vote

up vote
0
down vote

Guide:

• Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.




• What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

• It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

Guide:

• Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.




• What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

• It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

answered Jul 23 at 17:25

Siong Thye Goh

77.4k134795

77.4k134795

add a comment | 

add a comment | 

up vote
0
down vote

Only because the two sets of $X_1,X_2$ and $X_1,X_3$ does not guarantees you that either $X_1,X_2,X_3$ or $X_2,X_3$ are linear indenpent. Just consider the three vectors

$$X_1~=~beginpmatrix1\2\3endpmatrix X_2~=~beginpmatrix4\5\6endpmatrix X_3~=~beginpmatrix8\10\12endpmatrix$$

As you can see you restrictions are fullfilled but neither the set $X_2,X_3$ nor the whole set $X_1,X_2,X_3$ are linear independent.

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

add a comment | 

up vote
0
down vote

Only because the two sets of $X_1,X_2$ and $X_1,X_3$ does not guarantees you that either $X_1,X_2,X_3$ or $X_2,X_3$ are linear indenpent. Just consider the three vectors

$$X_1~=~beginpmatrix1\2\3endpmatrix X_2~=~beginpmatrix4\5\6endpmatrix X_3~=~beginpmatrix8\10\12endpmatrix$$

As you can see you restrictions are fullfilled but neither the set $X_2,X_3$ nor the whole set $X_1,X_2,X_3$ are linear independent.

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

add a comment | 

up vote
0
down vote

up vote
0
down vote

Only because the two sets of $X_1,X_2$ and $X_1,X_3$ does not guarantees you that either $X_1,X_2,X_3$ or $X_2,X_3$ are linear indenpent. Just consider the three vectors

$$X_1~=~beginpmatrix1\2\3endpmatrix X_2~=~beginpmatrix4\5\6endpmatrix X_3~=~beginpmatrix8\10\12endpmatrix$$

As you can see you restrictions are fullfilled but neither the set $X_2,X_3$ nor the whole set $X_1,X_2,X_3$ are linear independent.

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

Only because the two sets of $X_1,X_2$ and $X_1,X_3$ does not guarantees you that either $X_1,X_2,X_3$ or $X_2,X_3$ are linear indenpent. Just consider the three vectors

$$X_1~=~beginpmatrix1\2\3endpmatrix X_2~=~beginpmatrix4\5\6endpmatrix X_3~=~beginpmatrix8\10\12endpmatrix$$

As you can see you restrictions are fullfilled but neither the set $X_2,X_3$ nor the whole set $X_1,X_2,X_3$ are linear independent.

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 17:26

mrtaurho

750219

answered Jul 23 at 17:26

mrtaurho

750219

answered Jul 23 at 17:26

mrtaurho

750219

750219

add a comment | 

add a comment | 

 

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