Can someone explain what the exchange lemma in the following context is trying to say?
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Exchange lemma:
Let V be a vector space over F and let $v_1, v_2,...., v_n in V$. If
$$w in span(v_1,v_2,..,v_n) - span(v_2,....,v_n)$$
then, $$v_1 in span(w,v_2,....,v_n) - span(v_2,.....,v_n)$$
I have seen this lemma described in different ways, but I don't understand it as stated above. What does the author mean with the subtraction operation?. Is he thinking about a set operation? or is he thinking of a cross operation with subtraction?
linear-algebra
asked Aug 12 at 2:45
daniel
416312
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up vote
0
down vote
favorite
Exchange lemma:
Let V be a vector space over F and let $v_1, v_2,...., v_n in V$. If
$$w in span(v_1,v_2,..,v_n) - span(v_2,....,v_n)$$
then, $$v_1 in span(w,v_2,....,v_n) - span(v_2,.....,v_n)$$
I have seen this lemma described in different ways, but I don't understand it as stated above. What does the author mean with the subtraction operation?. Is he thinking about a set operation? or is he thinking of a cross operation with subtraction?
linear-algebra
asked Aug 12 at 2:45
daniel
416312
2
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Exchange lemma:
Let V be a vector space over F and let $v_1, v_2,...., v_n in V$. If
$$w in span(v_1,v_2,..,v_n) - span(v_2,....,v_n)$$
then, $$v_1 in span(w,v_2,....,v_n) - span(v_2,.....,v_n)$$
I have seen this lemma described in different ways, but I don't understand it as stated above. What does the author mean with the subtraction operation?. Is he thinking about a set operation? or is he thinking of a cross operation with subtraction?
linear-algebra
asked Aug 12 at 2:45
daniel
416312
Exchange lemma:
Let V be a vector space over F and let $v_1, v_2,...., v_n in V$. If
$$w in span(v_1,v_2,..,v_n) - span(v_2,....,v_n)$$
then, $$v_1 in span(w,v_2,....,v_n) - span(v_2,.....,v_n)$$
I have seen this lemma described in different ways, but I don't understand it as stated above. What does the author mean with the subtraction operation?. Is he thinking about a set operation? or is he thinking of a cross operation with subtraction?
linear-algebra
asked Aug 12 at 2:45
daniel
416312
asked Aug 12 at 2:45
daniel
416312
asked Aug 12 at 2:45
daniel
416312
asked Aug 12 at 2:45
daniel
416312
416312
2
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00
add a comment |Â
2
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00
2
2
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00
add a comment |Â
1 Answer
1
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0
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I think the idea here is that any
equivalence relation
on a set partitions that set into disjoint equivalence classes.
In the context of your question, we are given that
$, A := v_2,dots,v_n, ,$
a set of vectors in $ V. ,$
The equivalence relation is given by
$, x sim y iff xin textrmspan(y,A). ,$
Check that it satisfies the three properties of an equivalence relation
and that $, textrmspan(A) ,$ is an equivalence class.
What about other equivalence classes? If
$, w sim v_1 ,$ and $, w notin textrmspan(A) ,$
then by the symmetry property of $, sim ,$ we know that
$, v_1 sim w. ,$ Since equivalence classes are disjoint,
we know that $, v_1 notin textrmspan(A) ,$ also. That is exactly what the exchange lemma is stating.
answered Aug 12 at 14:08
Somos
11.7k11033
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I think the idea here is that any
equivalence relation
on a set partitions that set into disjoint equivalence classes.
In the context of your question, we are given that
$, A := v_2,dots,v_n, ,$
a set of vectors in $ V. ,$
The equivalence relation is given by
$, x sim y iff xin textrmspan(y,A). ,$
Check that it satisfies the three properties of an equivalence relation
and that $, textrmspan(A) ,$ is an equivalence class.
What about other equivalence classes? If
$, w sim v_1 ,$ and $, w notin textrmspan(A) ,$
then by the symmetry property of $, sim ,$ we know that
$, v_1 sim w. ,$ Since equivalence classes are disjoint,
we know that $, v_1 notin textrmspan(A) ,$ also. That is exactly what the exchange lemma is stating.
answered Aug 12 at 14:08
Somos
11.7k11033
add a comment |Â
up vote
0
down vote
I think the idea here is that any
equivalence relation
on a set partitions that set into disjoint equivalence classes.
In the context of your question, we are given that
$, A := v_2,dots,v_n, ,$
a set of vectors in $ V. ,$
The equivalence relation is given by
$, x sim y iff xin textrmspan(y,A). ,$
Check that it satisfies the three properties of an equivalence relation
and that $, textrmspan(A) ,$ is an equivalence class.
What about other equivalence classes? If
$, w sim v_1 ,$ and $, w notin textrmspan(A) ,$
then by the symmetry property of $, sim ,$ we know that
$, v_1 sim w. ,$ Since equivalence classes are disjoint,
we know that $, v_1 notin textrmspan(A) ,$ also. That is exactly what the exchange lemma is stating.
answered Aug 12 at 14:08
Somos
11.7k11033
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I think the idea here is that any
equivalence relation
on a set partitions that set into disjoint equivalence classes.
In the context of your question, we are given that
$, A := v_2,dots,v_n, ,$
a set of vectors in $ V. ,$
The equivalence relation is given by
$, x sim y iff xin textrmspan(y,A). ,$
Check that it satisfies the three properties of an equivalence relation
and that $, textrmspan(A) ,$ is an equivalence class.
What about other equivalence classes? If
$, w sim v_1 ,$ and $, w notin textrmspan(A) ,$
then by the symmetry property of $, sim ,$ we know that
$, v_1 sim w. ,$ Since equivalence classes are disjoint,
we know that $, v_1 notin textrmspan(A) ,$ also. That is exactly what the exchange lemma is stating.
answered Aug 12 at 14:08
Somos
11.7k11033
I think the idea here is that any
equivalence relation
on a set partitions that set into disjoint equivalence classes.
In the context of your question, we are given that
$, A := v_2,dots,v_n, ,$
a set of vectors in $ V. ,$
The equivalence relation is given by
$, x sim y iff xin textrmspan(y,A). ,$
Check that it satisfies the three properties of an equivalence relation
and that $, textrmspan(A) ,$ is an equivalence class.
What about other equivalence classes? If
$, w sim v_1 ,$ and $, w notin textrmspan(A) ,$
then by the symmetry property of $, sim ,$ we know that
$, v_1 sim w. ,$ Since equivalence classes are disjoint,
we know that $, v_1 notin textrmspan(A) ,$ also. That is exactly what the exchange lemma is stating.
answered Aug 12 at 14:08
Somos
11.7k11033
edited Aug 12 at 14:14
answered Aug 12 at 14:08
Somos
11.7k11033
answered Aug 12 at 14:08
Somos
11.7k11033
answered Aug 12 at 14:08
Somos
11.7k11033
11.7k11033
add a comment |Â
add a comment |Â
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2
This is the so-called "set difference" defined as follows: if $A$ and $B$ are sets, then $A setminus B := x mid x in A , textand , x notin B$.
– Branimir Ćaćić
Aug 12 at 2:52
If $w$ can be written as a linear combination of $v_1,v_2,dots,v_n$ but is unable to be written in such a way using only a linear combination of $v_2,dots,v_n$, then it follows that $v_1$ can be written as a linear combination of $w,v_2,dots,v_n$ but cannot be written as a linear combination using only $v_2,dots,v_n$.
– JMoravitz
Aug 12 at 3:00