Question 161324: What is a value of a where vektors are (2,0,a), (1,a,1) and (2,1,a) make basis for R^3?
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! Let's call the vectors u,v, and w.
u=(2,0,a)
v=(1,a,1)
w=(2,1,a)
For the set to be a basis, their dot products must be zero (orthogonality condition).
u*v=0
v*w=0
w*u=0
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1.u*v=2*1+0*a+a*1=0
1.
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2.v*w=1*2+a*1+1+a=0
2.
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3.w*u=2*2+1*0+a*a=0
3.
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From 1,
This value for a also needs to satisfy eq. 2 and eq. 3,
Checking with 2,
2+2a=0
2+2(-2)=-2
the value a does not solve 2 and you can check that it doesn't solve 3 either.
If the vectors formed a basis in R3, they must be orthogonal.
From 1,2, and 3, you cannot find an "a" that solves all three equations.
There is no solution.
Additionally, you could have also concluded this since eq. 3 has no real solution.
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