What does it mean for a vector to “be in the same translate” of a vector in a subspace?
$begingroup$
A question reads: consider the vector w = (2,3,9,1). Which (if any) of the following vectors: ... are in the same translate of V as w?
And the ellipsis is filling in for three other vectors.
This question is part b of a bigger question, the first part of which involved finding the matrix of a linear transformation (whose domain is R4 and codomain R2) given it's kernel (which is V). I'm also told in the question that V is a subspace of R4 and is spanned by two vectors (whose four elements are given).
I'm guessing this involves the transformation's matrix in some way, but I'm not sure on what "the same translate" means here. Could anyone offer some advice?
linear-algebra linear-transformations
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
$endgroup$
add a comment |
$begingroup$
A question reads: consider the vector w = (2,3,9,1). Which (if any) of the following vectors: ... are in the same translate of V as w?
And the ellipsis is filling in for three other vectors.
This question is part b of a bigger question, the first part of which involved finding the matrix of a linear transformation (whose domain is R4 and codomain R2) given it's kernel (which is V). I'm also told in the question that V is a subspace of R4 and is spanned by two vectors (whose four elements are given).
I'm guessing this involves the transformation's matrix in some way, but I'm not sure on what "the same translate" means here. Could anyone offer some advice?
linear-algebra linear-transformations
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
$endgroup$
$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53
add a comment |
$begingroup$
A question reads: consider the vector w = (2,3,9,1). Which (if any) of the following vectors: ... are in the same translate of V as w?
And the ellipsis is filling in for three other vectors.
This question is part b of a bigger question, the first part of which involved finding the matrix of a linear transformation (whose domain is R4 and codomain R2) given it's kernel (which is V). I'm also told in the question that V is a subspace of R4 and is spanned by two vectors (whose four elements are given).
I'm guessing this involves the transformation's matrix in some way, but I'm not sure on what "the same translate" means here. Could anyone offer some advice?
linear-algebra linear-transformations
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
$endgroup$
A question reads: consider the vector w = (2,3,9,1). Which (if any) of the following vectors: ... are in the same translate of V as w?
And the ellipsis is filling in for three other vectors.
This question is part b of a bigger question, the first part of which involved finding the matrix of a linear transformation (whose domain is R4 and codomain R2) given it's kernel (which is V). I'm also told in the question that V is a subspace of R4 and is spanned by two vectors (whose four elements are given).
I'm guessing this involves the transformation's matrix in some way, but I'm not sure on what "the same translate" means here. Could anyone offer some advice?
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
edited Dec 16 '18 at 16:53
James Ronald
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
asked Dec 16 '18 at 13:41
James RonaldJames Ronald
1257
1257
$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53
add a comment |
$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53
$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53
add a comment |
1 Answer
1
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oldest
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$begingroup$
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector.
This is shorthand for the set notation ${x + v mid v in V}.$
Note that $x$ itself is always a member of the translate $x + V,$
if $x$ is a member of a certain translate of $V$ then one of the ways to
name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$
that is, if the vector is named $u,$ then $u = w + v$ where $v in V.$
Now consider what happens when you apply the given transformation $L$ to
that vector: $L(w + v) = L(w) + L(v).$
You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
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add a comment |
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1 Answer
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active
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1 Answer
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oldest
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active
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votes
$begingroup$
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector.
This is shorthand for the set notation ${x + v mid v in V}.$
Note that $x$ itself is always a member of the translate $x + V,$
if $x$ is a member of a certain translate of $V$ then one of the ways to
name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$
that is, if the vector is named $u,$ then $u = w + v$ where $v in V.$
Now consider what happens when you apply the given transformation $L$ to
that vector: $L(w + v) = L(w) + L(v).$
You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
$endgroup$
add a comment |
$begingroup$
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector.
This is shorthand for the set notation ${x + v mid v in V}.$
Note that $x$ itself is always a member of the translate $x + V,$
if $x$ is a member of a certain translate of $V$ then one of the ways to
name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$
that is, if the vector is named $u,$ then $u = w + v$ where $v in V.$
Now consider what happens when you apply the given transformation $L$ to
that vector: $L(w + v) = L(w) + L(v).$
You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
$endgroup$
add a comment |
$begingroup$
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector.
This is shorthand for the set notation ${x + v mid v in V}.$
Note that $x$ itself is always a member of the translate $x + V,$
if $x$ is a member of a certain translate of $V$ then one of the ways to
name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$
that is, if the vector is named $u,$ then $u = w + v$ where $v in V.$
Now consider what happens when you apply the given transformation $L$ to
that vector: $L(w + v) = L(w) + L(v).$
You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
$endgroup$
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector.
This is shorthand for the set notation ${x + v mid v in V}.$
Note that $x$ itself is always a member of the translate $x + V,$
if $x$ is a member of a certain translate of $V$ then one of the ways to
name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$
that is, if the vector is named $u,$ then $u = w + v$ where $v in V.$
Now consider what happens when you apply the given transformation $L$ to
that vector: $L(w + v) = L(w) + L(v).$
You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
answered Dec 16 '18 at 20:31
David KDavid K
54.4k343120
54.4k343120
add a comment |
add a comment |
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$begingroup$
Is $V$ the kernel of the transformation?
$endgroup$
– David K
Dec 16 '18 at 16:06
$begingroup$
@DavidK yes it is, sorry should've mentioned that
$endgroup$
– James Ronald
Dec 16 '18 at 16:53