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Linear independence/dependence and Span [closed]
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I am not entirely sure about this concept. Suppose that we have two sets of vectors say $a,b,c$ and $u,v,w$. The span of these two vector sets are equal to each other. If one set of vectors $a,b,c$ is linearly independent/or dependent will the other set of vectors u,v,w be linearly independent/or dependent i.e if one is linear independent will the other be linear dependent if we know their spans equal?
linear-algebra
edited Aug 2 at 16:35
Javi
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asked Aug 2 at 15:58
Molly
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closed as off-topic by amWhy, Simply Beautiful Art, José Carlos Santos, Isaac Browne, Adrian Keister Aug 3 at 0:04
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Simply Beautiful Art, Jos̩ Carlos Santos, Isaac Browne, Adrian Keister
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up vote
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I am not entirely sure about this concept. Suppose that we have two sets of vectors say $a,b,c$ and $u,v,w$. The span of these two vector sets are equal to each other. If one set of vectors $a,b,c$ is linearly independent/or dependent will the other set of vectors u,v,w be linearly independent/or dependent i.e if one is linear independent will the other be linear dependent if we know their spans equal?
linear-algebra
edited Aug 2 at 16:35
Javi
2,1481625
asked Aug 2 at 15:58
Molly
303
closed as off-topic by amWhy, Simply Beautiful Art, José Carlos Santos, Isaac Browne, Adrian Keister Aug 3 at 0:04
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Simply Beautiful Art, Jos̩ Carlos Santos, Isaac Browne, Adrian Keister
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am not entirely sure about this concept. Suppose that we have two sets of vectors say $a,b,c$ and $u,v,w$. The span of these two vector sets are equal to each other. If one set of vectors $a,b,c$ is linearly independent/or dependent will the other set of vectors u,v,w be linearly independent/or dependent i.e if one is linear independent will the other be linear dependent if we know their spans equal?
linear-algebra
edited Aug 2 at 16:35
Javi
2,1481625
asked Aug 2 at 15:58
Molly
303
I am not entirely sure about this concept. Suppose that we have two sets of vectors say $a,b,c$ and $u,v,w$. The span of these two vector sets are equal to each other. If one set of vectors $a,b,c$ is linearly independent/or dependent will the other set of vectors u,v,w be linearly independent/or dependent i.e if one is linear independent will the other be linear dependent if we know their spans equal?
linear-algebra
edited Aug 2 at 16:35
Javi
2,1481625
asked Aug 2 at 15:58
Molly
303
edited Aug 2 at 16:35
Javi
2,1481625
edited Aug 2 at 16:35
Javi
2,1481625
edited Aug 2 at 16:35
Javi
2,1481625
2,1481625
asked Aug 2 at 15:58
Molly
303
asked Aug 2 at 15:58
Molly
303
asked Aug 2 at 15:58
Molly
303
303
closed as off-topic by amWhy, Simply Beautiful Art, José Carlos Santos, Isaac Browne, Adrian Keister Aug 3 at 0:04
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Simply Beautiful Art, Jos̩ Carlos Santos, Isaac Browne, Adrian Keister
closed as off-topic by amWhy, Simply Beautiful Art, José Carlos Santos, Isaac Browne, Adrian Keister Aug 3 at 0:04
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Simply Beautiful Art, Jos̩ Carlos Santos, Isaac Browne, Adrian Keister
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2 Answers
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If $a,b,c$ is a L.I. set, then $textdim,textspana,b,c = 3$, and therefore since $textspana,b,c=textspanu,v,w$, we must have that $textdim,textspanu,v,w = 3$.
Now since there are only 3 elements in $u,v,w$ and $textdim,textspanu,v,w = 3$, we must have that $u,v,w$ is a L.I. set also.
answered Aug 2 at 16:04
bobcliffe
435
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Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n geq 3$ (sizes of the vectors), we have the following cases:
Cases:
If $dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.
If $dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.
If $dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.
$dim S$ could not be greater than $3$ since we only have three vectors
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
If $a,b,c$ is a L.I. set, then $textdim,textspana,b,c = 3$, and therefore since $textspana,b,c=textspanu,v,w$, we must have that $textdim,textspanu,v,w = 3$.
Now since there are only 3 elements in $u,v,w$ and $textdim,textspanu,v,w = 3$, we must have that $u,v,w$ is a L.I. set also.
answered Aug 2 at 16:04
bobcliffe
435
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up vote
4
down vote
If $a,b,c$ is a L.I. set, then $textdim,textspana,b,c = 3$, and therefore since $textspana,b,c=textspanu,v,w$, we must have that $textdim,textspanu,v,w = 3$.
Now since there are only 3 elements in $u,v,w$ and $textdim,textspanu,v,w = 3$, we must have that $u,v,w$ is a L.I. set also.
answered Aug 2 at 16:04
bobcliffe
435
add a comment |Â
up vote
4
down vote
up vote
4
down vote
If $a,b,c$ is a L.I. set, then $textdim,textspana,b,c = 3$, and therefore since $textspana,b,c=textspanu,v,w$, we must have that $textdim,textspanu,v,w = 3$.
Now since there are only 3 elements in $u,v,w$ and $textdim,textspanu,v,w = 3$, we must have that $u,v,w$ is a L.I. set also.
answered Aug 2 at 16:04
bobcliffe
435
If $a,b,c$ is a L.I. set, then $textdim,textspana,b,c = 3$, and therefore since $textspana,b,c=textspanu,v,w$, we must have that $textdim,textspanu,v,w = 3$.
Now since there are only 3 elements in $u,v,w$ and $textdim,textspanu,v,w = 3$, we must have that $u,v,w$ is a L.I. set also.
answered Aug 2 at 16:04
bobcliffe
435
answered Aug 2 at 16:04
bobcliffe
435
answered Aug 2 at 16:04
bobcliffe
435
answered Aug 2 at 16:04
bobcliffe
435
435
add a comment |Â
add a comment |Â
up vote
0
down vote
Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n geq 3$ (sizes of the vectors), we have the following cases:
Cases:
If $dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.
If $dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.
If $dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.
$dim S$ could not be greater than $3$ since we only have three vectors
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
add a comment |Â
up vote
0
down vote
Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n geq 3$ (sizes of the vectors), we have the following cases:
Cases:
If $dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.
If $dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.
If $dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.
$dim S$ could not be greater than $3$ since we only have three vectors
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n geq 3$ (sizes of the vectors), we have the following cases:
Cases:
If $dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.
If $dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.
If $dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.
$dim S$ could not be greater than $3$ since we only have three vectors
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n geq 3$ (sizes of the vectors), we have the following cases:
Cases:
If $dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.
If $dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.
If $dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.
$dim S$ could not be greater than $3$ since we only have three vectors
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
answered Aug 2 at 16:07
Ahmad Bazzi
2,172417
2,172417
add a comment |Â
add a comment |Â