Find a basis of the subspaces of R4 generated by the vectors v1=(1,1,2,0),v2=(−1,0,1,0),v3=(2,−2,0,0),v4=(0,0,−1,2)
First of all I wrote these vectors as rows of a matrix then applied the following transformations to reduce the matrix in row echelon form:-
R2+R1 & R3−2R1
Then R3+4R2 and finally 110R3
Then I wrote the non zero rows in row echelon form as B=(1,1,2,0),(0,1,3,0),(0,0,1,0) which forms basis .
Am I right here?
Answer
Using the vectors given in the problem, we can define a set: A={v1,v2,v3,v4}
Remember the definition of a Basis set:
Given: V is a vector space and B is a subset of V, we say that B is a basis of V IFF V=span(B) and B is linearly independent.
Row-Reducing the matrix form of the set A yields:
[1−12010−20210−10002]
<=>
[1000010000100001]
Since each column contains a leading "1", we can say that the set A is linearly independent.
This result implies that the set A itself satisfies the definition of a basis.
B={v1,v2,v3,v4}
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