Mixing
Finding the orthogonal complement of a subspace of $mathbb{R}^4$
$begingroup$
Let $V$ be the vector space ${(x,y,z,w)inmathbb{R}^4:x+y-z=0 text{and} x+y+w=0}$. Then, what would be a basis for the orthogonal complement of $V$.
I think we have to find the null space of the column space spanned by the above two vectors, i.e, the null space of
$$begin{pmatrix}1&1&0&0\1&1&0&0\2&0&0&0\0&-2&0&0end{pmatrix}$$
Am I right? Thanks beforehand.
linear-algebra matrices vector-spaces orthogonality
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
$endgroup$
add a comment |
$begingroup$
Let $V$ be the vector space ${(x,y,z,w)inmathbb{R}^4:x+y-z=0 text{and} x+y+w=0}$. Then, what would be a basis for the orthogonal complement of $V$.
I think we have to find the null space of the column space spanned by the above two vectors, i.e, the null space of
$$begin{pmatrix}1&1&0&0\1&1&0&0\2&0&0&0\0&-2&0&0end{pmatrix}$$
Am I right? Thanks beforehand.
linear-algebra matrices vector-spaces orthogonality
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
$endgroup$
$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37
add a comment |
$begingroup$
Let $V$ be the vector space ${(x,y,z,w)inmathbb{R}^4:x+y-z=0 text{and} x+y+w=0}$. Then, what would be a basis for the orthogonal complement of $V$.
I think we have to find the null space of the column space spanned by the above two vectors, i.e, the null space of
$$begin{pmatrix}1&1&0&0\1&1&0&0\2&0&0&0\0&-2&0&0end{pmatrix}$$
Am I right? Thanks beforehand.
linear-algebra matrices vector-spaces orthogonality
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
$endgroup$
Let $V$ be the vector space ${(x,y,z,w)inmathbb{R}^4:x+y-z=0 text{and} x+y+w=0}$. Then, what would be a basis for the orthogonal complement of $V$.
I think we have to find the null space of the column space spanned by the above two vectors, i.e, the null space of
$$begin{pmatrix}1&1&0&0\1&1&0&0\2&0&0&0\0&-2&0&0end{pmatrix}$$
Am I right? Thanks beforehand.
linear-algebra matrices vector-spaces orthogonality
linear-algebra matrices vector-spaces orthogonality
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
edited Jan 25 at 9:40
José Carlos Santos
168k23132236
168k23132236
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
asked Jan 25 at 9:31
vidyarthividyarthi
3,0361833
3,0361833
$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37
add a comment |
$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37
$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The space $V$ is the space of those vectors $vinmathbb{R}^4$ such that $bigllangle v,(1,1,-1,0)bigrrangle=bigllangle v,(1,1,0,1)bigrrangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
$endgroup$
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
|
show 1 more comment
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The space $V$ is the space of those vectors $vinmathbb{R}^4$ such that $bigllangle v,(1,1,-1,0)bigrrangle=bigllangle v,(1,1,0,1)bigrrangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
$endgroup$
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
|
show 1 more comment
$begingroup$
The space $V$ is the space of those vectors $vinmathbb{R}^4$ such that $bigllangle v,(1,1,-1,0)bigrrangle=bigllangle v,(1,1,0,1)bigrrangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
$endgroup$
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
|
show 1 more comment
$begingroup$
The space $V$ is the space of those vectors $vinmathbb{R}^4$ such that $bigllangle v,(1,1,-1,0)bigrrangle=bigllangle v,(1,1,0,1)bigrrangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
$endgroup$
The space $V$ is the space of those vectors $vinmathbb{R}^4$ such that $bigllangle v,(1,1,-1,0)bigrrangle=bigllangle v,(1,1,0,1)bigrrangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
answered Jan 25 at 9:38
José Carlos SantosJosé Carlos Santos
168k23132236
168k23132236
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
|
show 1 more comment
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
how did you arrive at that?
$endgroup$
– vidyarthi
Jan 25 at 9:39
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
At what? Are you talking about my first sentence or the second one?
$endgroup$
– José Carlos Santos
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
I meant, what is $v$ in this case?
$endgroup$
– vidyarthi
Jan 25 at 9:40
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
The vector $v$ is a vector of the form $(x,y,z,w)$, with $x,y,z,winmathbb R$.
$endgroup$
– José Carlos Santos
Jan 25 at 9:41
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
$begingroup$
thanks! so just remove the variables out and form the dot product, right? Quite easy!
$endgroup$
– vidyarthi
Jan 25 at 9:42
|
show 1 more comment
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$begingroup$
What is $t$ in the definition of $V$?
$endgroup$
– José Carlos Santos
Jan 25 at 9:36
$begingroup$
@JoséCarlosSantos thanks again! edited
$endgroup$
– vidyarthi
Jan 25 at 9:37