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Rakesh Kumar posted an Question
August 10, 2020 • 05:00 am
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30 points
30 points
- IIT JAM
- Mathematics (MA)
V let fr° r be the linear mapping defined by then which of the tollowing no (1,1,1,1) and (-1.0,1.2) torms a basis of ker f (a) (b) (2,1.-1,0) and (1,2,0.1) for
V Let FR° R be the linear mapping defined by then which of the tollowing nO (1,1,1,1) and (-1.0,1.2) torms a basis of Ker F (A) (B) (2,1.-1,0) and (1,2,0.1) forms a basis of Ker F (C) (0.0,0,1) form a basis of Ker F (D) None
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Shashi ranjan sinha
see this linear map... C is also true
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Rakesh kumar
according to me only a& b may be true
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Rakesh kumar
option c to wrong hoga because it is not generated
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why?
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(0,0,0,1) non zero vector hai to linear independent hai aur iske through jo subspace generate hoga ....uska ye basis hoga....
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according to definition of basis
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Bhai basis linearly lidependent hota h
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dependent nahi
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single non zero vector is always Linear independent
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(0,0,0,1) is a non zero vector in R⁴
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ha li to h
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ji bilkul
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Kaya generate Kar rha h
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Rakesh kumar
nothing is defined
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Shashi ranjan sinha
since f is not given, therefore the options 1,2,3 can hold... they are in accordance with rank nullity theorem...see the attachment
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if it is a basis so li as well as generat
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yeah ask your doubt
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Yes of course
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Consider option a.... if the given vectors are basis of kernel of F which is a subspace of R⁴, so it can be extended to form a basis of R⁴....if u map the other two vectors in the extended basis to any arbitrary non zero vectors in R³, then that map is a linear mapping and call it as your F
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in this way , with given basis of kernel of f, u can define ur linear map f
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see this theorem
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Shashi ranjan sinha
I just want to make sure that you understand this question.... but if still have any doubt, I will try to clarify it
Bhai one element tabhi basis hoga jab o set ke sare element ko generate karega; li to h but generate nahi Kar rha h
basis of Kernel bol raha hai... basis of R⁴ nahi
aap bilkul sahi hain ki basis saare elements ko generate karta hai....but jis space ka wo basis hai uske hi saare vectors ko generate karega , aur kisi vectors ko nhi