What does “Reflection along the subspace generated by v” means?
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I got a problem which includes "Reflection along the subspace generated by $v$ in $mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{bot}$?
linear-algebra vectors euclidean-geometry reflection
asked Nov 8 at 4:19
PSG
1908
|
show 6 more comments
up vote
0
down vote
favorite
I got a problem which includes "Reflection along the subspace generated by $v$ in $mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{bot}$?
linear-algebra vectors euclidean-geometry reflection
asked Nov 8 at 4:19
PSG
1908
I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
1
I have written it down.
– PSG
9 hours ago
|
show 6 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I got a problem which includes "Reflection along the subspace generated by $v$ in $mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{bot}$?
linear-algebra vectors euclidean-geometry reflection
asked Nov 8 at 4:19
PSG
1908
I got a problem which includes "Reflection along the subspace generated by $v$ in $mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{bot}$?
linear-algebra vectors euclidean-geometry reflection
linear-algebra vectors euclidean-geometry reflection
asked Nov 8 at 4:19
PSG
1908
asked Nov 8 at 4:19
PSG
1908
asked Nov 8 at 4:19
PSG
1908
asked Nov 8 at 4:19
PSG
1908
asked Nov 8 at 4:19
PSG
1908
1908
I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
1
I have written it down.
– PSG
9 hours ago
|
show 6 more comments
I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
1
I have written it down.
– PSG
9 hours ago
I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
1
1
I have written it down.
– PSG
9 hours ago
I have written it down.
– PSG
9 hours ago
|
show 6 more comments
1 Answer
1
active
oldest
votes
up vote
1
down vote
Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:
let $mathbf{r}$ be the reflection of the original vector $mathbf{x}$, along the subspace generated by $mathbf{v}$. Then both $mathbf{r}$ and $mathbf{x}$ has same projection onto $mathbf{v}$ as reflection along $mathbf{v}$ will keep the projection onto $mathbf{v}$ intact.
So, $mathbf{x}+mathbf{r}= 2 times $ (Projection of $mathbf{x}$ onto the subspace of scalar multiples of $mathbf{v})= {2 x cdot vover |v|^2}vRightarrow mathbf{r}={2 x cdot vover |v|^2}v- mathbf{x}$
answered 13 hours ago
PSG
1908
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:
let $mathbf{r}$ be the reflection of the original vector $mathbf{x}$, along the subspace generated by $mathbf{v}$. Then both $mathbf{r}$ and $mathbf{x}$ has same projection onto $mathbf{v}$ as reflection along $mathbf{v}$ will keep the projection onto $mathbf{v}$ intact.
So, $mathbf{x}+mathbf{r}= 2 times $ (Projection of $mathbf{x}$ onto the subspace of scalar multiples of $mathbf{v})= {2 x cdot vover |v|^2}vRightarrow mathbf{r}={2 x cdot vover |v|^2}v- mathbf{x}$
answered 13 hours ago
PSG
1908
add a comment |
up vote
1
down vote
Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:
let $mathbf{r}$ be the reflection of the original vector $mathbf{x}$, along the subspace generated by $mathbf{v}$. Then both $mathbf{r}$ and $mathbf{x}$ has same projection onto $mathbf{v}$ as reflection along $mathbf{v}$ will keep the projection onto $mathbf{v}$ intact.
So, $mathbf{x}+mathbf{r}= 2 times $ (Projection of $mathbf{x}$ onto the subspace of scalar multiples of $mathbf{v})= {2 x cdot vover |v|^2}vRightarrow mathbf{r}={2 x cdot vover |v|^2}v- mathbf{x}$
answered 13 hours ago
PSG
1908
add a comment |
up vote
1
down vote
up vote
1
down vote
Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:
let $mathbf{r}$ be the reflection of the original vector $mathbf{x}$, along the subspace generated by $mathbf{v}$. Then both $mathbf{r}$ and $mathbf{x}$ has same projection onto $mathbf{v}$ as reflection along $mathbf{v}$ will keep the projection onto $mathbf{v}$ intact.
So, $mathbf{x}+mathbf{r}= 2 times $ (Projection of $mathbf{x}$ onto the subspace of scalar multiples of $mathbf{v})= {2 x cdot vover |v|^2}vRightarrow mathbf{r}={2 x cdot vover |v|^2}v- mathbf{x}$
answered 13 hours ago
PSG
1908
Thanks to Gerry Myerson who guided me to solve this problem. Here is the solution:
let $mathbf{r}$ be the reflection of the original vector $mathbf{x}$, along the subspace generated by $mathbf{v}$. Then both $mathbf{r}$ and $mathbf{x}$ has same projection onto $mathbf{v}$ as reflection along $mathbf{v}$ will keep the projection onto $mathbf{v}$ intact.
So, $mathbf{x}+mathbf{r}= 2 times $ (Projection of $mathbf{x}$ onto the subspace of scalar multiples of $mathbf{v})= {2 x cdot vover |v|^2}vRightarrow mathbf{r}={2 x cdot vover |v|^2}v- mathbf{x}$
answered 13 hours ago
PSG
1908
edited 12 hours ago
answered 13 hours ago
PSG
1908
answered 13 hours ago
PSG
1908
answered 13 hours ago
PSG
1908
1908
add a comment |
add a comment |
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I would interpret it as reflection in the one-dimensional subspace consisting of the scalar multiples of $v$. But first I'd read the rest of whatever the problem says, and see whether I could deduce the meaning from context. And if I couldn't, then I would ask the person who gave me the problem what interpretation she would give.
– Gerry Myerson
Nov 8 at 5:31
Anything to say, PSG?
– Gerry Myerson
Nov 13 at 10:19
Well I think it's as you said in the first comment. Reflection in the 1d subspace. To solve the problem I need it continuous, better if we can get a formula
– PSG
Nov 13 at 10:34
Do you know how to compute the projection of a vector onto the subspace of scalar multiples of $v$?
– Gerry Myerson
Nov 13 at 11:21
1
I have written it down.
– PSG
9 hours ago