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If the images of vectors are linearly dependent, then they are linearly dependent [duplicate]
This question already has an answer here:
Can linearly independent vectors have linearly dependent images?
2 answers
I know that if the images of vectors are linearly independent, then the vectors are linearly independent. But will the statement still hold if we change independent to dependent? I tried testing with common linear transformations and so far the case holds. Are there any counter examples to it?
linear-algebra linear-transformations
asked Nov 21 '18 at 19:39
user161872user161872
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marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, Kelvin Lois, José Carlos Santos linear-algebra
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Nov 22 '18 at 8:28
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Can linearly independent vectors have linearly dependent images?
2 answers
I know that if the images of vectors are linearly independent, then the vectors are linearly independent. But will the statement still hold if we change independent to dependent? I tried testing with common linear transformations and so far the case holds. Are there any counter examples to it?
linear-algebra linear-transformations
asked Nov 21 '18 at 19:39
user161872user161872
1
marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, Kelvin Lois, José Carlos Santos linear-algebra
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Nov 22 '18 at 8:28
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43
add a comment |
This question already has an answer here:
Can linearly independent vectors have linearly dependent images?
2 answers
I know that if the images of vectors are linearly independent, then the vectors are linearly independent. But will the statement still hold if we change independent to dependent? I tried testing with common linear transformations and so far the case holds. Are there any counter examples to it?
linear-algebra linear-transformations
asked Nov 21 '18 at 19:39
user161872user161872
1
This question already has an answer here:
Can linearly independent vectors have linearly dependent images?
2 answers
I know that if the images of vectors are linearly independent, then the vectors are linearly independent. But will the statement still hold if we change independent to dependent? I tried testing with common linear transformations and so far the case holds. Are there any counter examples to it?
This question already has an answer here:
Can linearly independent vectors have linearly dependent images?
2 answers
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Nov 21 '18 at 19:39
user161872user161872
1
asked Nov 21 '18 at 19:39
user161872user161872
1
asked Nov 21 '18 at 19:39
user161872user161872
1
asked Nov 21 '18 at 19:39
user161872user161872
1
asked Nov 21 '18 at 19:39
user161872user161872
1
1
marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, Kelvin Lois, José Carlos Santos linear-algebra
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Nov 22 '18 at 8:28
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, Kelvin Lois, José Carlos Santos linear-algebra
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Nov 22 '18 at 8:28
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43
add a comment |
3
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43
3
3
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43
add a comment |
1 Answer
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No. Take $f:mathbb Rtomathbb R$ given by $f(x)=0$. Then ${f(1)} = {0}$ is linearly dependent, but ${1}$ is linearly independent.
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
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1 Answer
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1 Answer
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No. Take $f:mathbb Rtomathbb R$ given by $f(x)=0$. Then ${f(1)} = {0}$ is linearly dependent, but ${1}$ is linearly independent.
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
add a comment |
No. Take $f:mathbb Rtomathbb R$ given by $f(x)=0$. Then ${f(1)} = {0}$ is linearly dependent, but ${1}$ is linearly independent.
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
add a comment |
No. Take $f:mathbb Rtomathbb R$ given by $f(x)=0$. Then ${f(1)} = {0}$ is linearly dependent, but ${1}$ is linearly independent.
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
No. Take $f:mathbb Rtomathbb R$ given by $f(x)=0$. Then ${f(1)} = {0}$ is linearly dependent, but ${1}$ is linearly independent.
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
answered Nov 21 '18 at 19:43
FedericoFederico
4,829514
4,829514
add a comment |
add a comment |
3
Try a projection.
– Ramiro de la Vega
Nov 21 '18 at 19:43