Finding a basis for the range space

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    Basis Range Space
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The dimension of the vector space V is established as 3, but there is confusion regarding the basis provided, which includes the vectors (1,1,1,2), (1,2,-3,1), and (3,4,-1,5). It is argued that these vectors may not be linearly independent, thus questioning their validity as a basis for the column space of the matrix. The discussion highlights that any three linearly dependent vectors cannot serve as a basis, suggesting the need to select different columns for an independent set. Ultimately, the participant expresses understanding of how to proceed with the problem. The importance of verifying linear independence in determining a valid basis is emphasized.
Janiceleong26
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1. Homework Statement
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I've found the dimension of V to be 3.
According to the solutions, it seems that the basis can be written straight away, { (1,1,1,2), (1,2,-3,1), (3,4,-1,5) } (which is also the basis for the column space of the matrix), without verifying the vectors are linearly independent.. how come? The vectors in the matrix are not necessary linearly independent..
 
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Those three vectors cannot be a basis of anything, because they are linearly dependent. You can choose any three other columns to get an independent set.
 
mfb said:
Those three vectors cannot be a basis of anything, because they are linearly dependent. You can choose any three other columns to get an independent set.
Oh right.. I know how to do already, thanks!
 

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