Question 31476: Let S={v1, v2, v3}where v1^T=[1 0] v2^T=[0 1] v3^T=[-1 1]
Find every subset of S that is a basis for R^2.
I have the matrix
[1 0 -1 0]
[0 1 1 0]
where x1-x3=0 and x2+x3=o and one basis of {v1, v2} but I don't know where to go from there.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Let S={v1, v2, v3}where v1^T=[1 0] v2^T=[0 1] v3^T=[-1 1]
Find every subset of S that is a basis for R^2.
BASIS DEMANDS 2 THINGS...
1. THEY SHOULD BE LINMEARLY INDEPENDENT
2.THEY SHOULD SPAN THE ENTIRE SPACE.
FOR FOR 2 DIMENSIONS (R^2)THERE WILL BE 2 AND ONLY 2 INDEPENDENT VECTORS WHICH SATISFY BOTH THE ABOVE REQUIREMENTS.HENCE WE NEED TO FIND ONLY SUBSETS OF 2 ELEMENTS...THEY COULD BE
1...V1V2....OR.....2..V1V3....OR.....3...V2V3
FOR TESTING INDEPENTNESS AND SPANNING THE ENTIRE SPACE HERE WE ONLY NEED TO CHECK INDEPENTNESS AS THEN IN R^2 , 2 INDEPENDENT VECTORS WILL CERAINLY SPAN THE ENTIRE SPACE.
1.V1V2,,,
LET AV1+BV2=0
A(1,0)+B(0,1)=0
THIS CAN HAPPEN ONLY WHEN A=B=0..SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.
2.V1V3
A(1,0)+B(-1,1)=0...THIS ALSO GIVES A=B=0...SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.
3.V2V3
A(0,1)+B(-1,1)=0.....ALSO GIVES A=B=0...SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.
SO ALL 3 SUBSETS CAN FORM A BASIS IN R^2
I have the matrix
[1 0 -1 0]
[0 1 1 0]
where x1-x3=0 and x2+x3=o and one basis of {v1, v2} but I don't know where to go from there.
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