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If u is orthogonal to both v and w, and u not equal to 0, argue that u is not in the span of v and w. (

2

$begingroup$

QN: If u is orthogonal to both v and w, and u ≠ 0, argue that u is not in the span of v and w.

Where I am at:

I get stuck when it comes to solving my augmented matrix with Gauss Jordan Elimination.

I also tried formulating the following steps to solve the problem.

1. Create instances of u, v and w that pertain to the question. My visualisation in Geogebra can be viewed here: https://ggbm.at/b6xvwhpa




2. Set u = av + bw = u (where a and b are constants)



3. Disprove (2)

However, I could not get past step 1.

Any pointers would be greatly appreciated.

linear-algebra matrices

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

add a comment | 

2

$begingroup$

QN: If u is orthogonal to both v and w, and u ≠ 0, argue that u is not in the span of v and w.

Where I am at:

I get stuck when it comes to solving my augmented matrix with Gauss Jordan Elimination.

I also tried formulating the following steps to solve the problem.

1. Create instances of u, v and w that pertain to the question. My visualisation in Geogebra can be viewed here: https://ggbm.at/b6xvwhpa




2. Set u = av + bw = u (where a and b are constants)



3. Disprove (2)

However, I could not get past step 1.

Any pointers would be greatly appreciated.

linear-algebra matrices

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

QN: If u is orthogonal to both v and w, and u ≠ 0, argue that u is not in the span of v and w.

Where I am at:

I get stuck when it comes to solving my augmented matrix with Gauss Jordan Elimination.

I also tried formulating the following steps to solve the problem.

1. Create instances of u, v and w that pertain to the question. My visualisation in Geogebra can be viewed here: https://ggbm.at/b6xvwhpa




2. Set u = av + bw = u (where a and b are constants)



3. Disprove (2)

However, I could not get past step 1.

Any pointers would be greatly appreciated.

linear-algebra matrices

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited 1 hour ago

YuiTo Cheng

2,52341037

edited 1 hour ago

YuiTo Cheng

2,52341037

asked 1 hour ago

Dimen3ionalDimen3ional

132

New contributor

Dimen3ional is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 1 hour ago

Dimen3ionalDimen3ional

132

2 Answers
2

active

oldest

votes

4

$begingroup$

$defmyvec#1bf#1$
This is the same as saying

if $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, and $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=myvec 0$.

So, if $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=amyvec v+bmyvec w$ for some scalars $a,b$. Assuming also $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, this means $myvec ucdot myvec v=0$ and $myvec ucdotmyvec w=0$, so
$$myvec ucdot myvec u=myvec ucdot(amyvec v+bmyvec w)=a(myvec ucdot myvec v)+b(myvec ucdot myvec w)=a0+b0=0 .$$
Since $myvec ucdot myvec u=0$ we have $myvec u=myvec 0$.

share|cite|improve this answer

answered 1 hour ago

DavidDavid

70.1k668131

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I thought about this, however, the question specifically states that u can not be zero. If u can't be zero does that not prevent the above proof, or am I looking at this the wrong way?
  $endgroup$
  – Dimen3ional
  59 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you studied proof by contradiction or by contrapositive? That's what this is.
  $endgroup$
  – David
  58 mins ago
  
  

  
  
  
  

add a comment | 

2

$begingroup$

If $u$ belongs to the span of $v$ and $w$ the $u=av+bw$ for some scalars $a$ and $b$. Since $langle u, v rangle=0$ and $langle u, w rangle=0$ we get $langle u, (av+bw) rangle=0$ so $langle u, u rangle=0$. This means $u=0$ which is a contradiction.

share|cite|improve this answer

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

$endgroup$

add a comment | 

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4

$begingroup$

$defmyvec#1bf#1$
This is the same as saying

if $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, and $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=myvec 0$.

So, if $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=amyvec v+bmyvec w$ for some scalars $a,b$. Assuming also $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, this means $myvec ucdot myvec v=0$ and $myvec ucdotmyvec w=0$, so
$$myvec ucdot myvec u=myvec ucdot(amyvec v+bmyvec w)=a(myvec ucdot myvec v)+b(myvec ucdot myvec w)=a0+b0=0 .$$
Since $myvec ucdot myvec u=0$ we have $myvec u=myvec 0$.

share|cite|improve this answer

answered 1 hour ago

DavidDavid

70.1k668131

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I thought about this, however, the question specifically states that u can not be zero. If u can't be zero does that not prevent the above proof, or am I looking at this the wrong way?
  $endgroup$
  – Dimen3ional
  59 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you studied proof by contradiction or by contrapositive? That's what this is.
  $endgroup$
  – David
  58 mins ago
  
  

  
  
  
  

add a comment | 

4

$begingroup$

$defmyvec#1bf#1$
This is the same as saying

if $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, and $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=myvec 0$.

So, if $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=amyvec v+bmyvec w$ for some scalars $a,b$. Assuming also $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, this means $myvec ucdot myvec v=0$ and $myvec ucdotmyvec w=0$, so
$$myvec ucdot myvec u=myvec ucdot(amyvec v+bmyvec w)=a(myvec ucdot myvec v)+b(myvec ucdot myvec w)=a0+b0=0 .$$
Since $myvec ucdot myvec u=0$ we have $myvec u=myvec 0$.

share|cite|improve this answer

answered 1 hour ago

DavidDavid

70.1k668131

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I thought about this, however, the question specifically states that u can not be zero. If u can't be zero does that not prevent the above proof, or am I looking at this the wrong way?
  $endgroup$
  – Dimen3ional
  59 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you studied proof by contradiction or by contrapositive? That's what this is.
  $endgroup$
  – David
  58 mins ago
  
  

  
  
  
  

add a comment | 

$begingroup$

$defmyvec#1bf#1$
This is the same as saying

if $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, and $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=myvec 0$.

So, if $myvec u$ is in the span of $myvec v$ and $myvec w$, then $myvec u=amyvec v+bmyvec w$ for some scalars $a,b$. Assuming also $myvec u$ is orthogonal to both $myvec v$ and $myvec w$, this means $myvec ucdot myvec v=0$ and $myvec ucdotmyvec w=0$, so
$$myvec ucdot myvec u=myvec ucdot(amyvec v+bmyvec w)=a(myvec ucdot myvec v)+b(myvec ucdot myvec w)=a0+b0=0 .$$
Since $myvec ucdot myvec u=0$ we have $myvec u=myvec 0$.

share|cite|improve this answer

answered 1 hour ago

DavidDavid

70.1k668131

$endgroup$

share|cite|improve this answer

answered 1 hour ago

DavidDavid

70.1k668131

answered 1 hour ago

DavidDavid

70.1k668131

answered 1 hour ago

DavidDavid

70.1k668131

2

$begingroup$

If $u$ belongs to the span of $v$ and $w$ the $u=av+bw$ for some scalars $a$ and $b$. Since $langle u, v rangle=0$ and $langle u, w rangle=0$ we get $langle u, (av+bw) rangle=0$ so $langle u, u rangle=0$. This means $u=0$ which is a contradiction.

share|cite|improve this answer

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

$endgroup$

add a comment | 

2

$begingroup$

If $u$ belongs to the span of $v$ and $w$ the $u=av+bw$ for some scalars $a$ and $b$. Since $langle u, v rangle=0$ and $langle u, w rangle=0$ we get $langle u, (av+bw) rangle=0$ so $langle u, u rangle=0$. This means $u=0$ which is a contradiction.

share|cite|improve this answer

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

$endgroup$

add a comment | 

$begingroup$

If $u$ belongs to the span of $v$ and $w$ the $u=av+bw$ for some scalars $a$ and $b$. Since $langle u, v rangle=0$ and $langle u, w rangle=0$ we get $langle u, (av+bw) rangle=0$ so $langle u, u rangle=0$. This means $u=0$ which is a contradiction.

share|cite|improve this answer

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

$endgroup$

share|cite|improve this answer

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270

answered 1 hour ago

Kavi Rama MurthyKavi Rama Murthy

75.6k53270