Construct an orthonormal basis for the subspace of R3 spanned by the vectors (1,-1,1)
and (2,0,4)
1
Expert's answer
2021-08-17T17:50:48-0400
Let us find a∈R such that the vectors u=(1,−1,1) and v=(1,−1,1)+a(2,0,4)=(1+2a,−1,1+4a) are orthogonal. Then the dot product of these vectors is equal to 0, that is 1+2a+1+1+4a=0. It follows that 6a=−3, and hence a=−21.
Then v=(1+2(−21),−1,1+4(−21))=(0,−1,−1). Since ∣u∣=1+1+1=3 and ∣u∣=0+1+1=2, we conclude that the vectors ∣u∣u=(31,−31,31) and ∣v∣v=(0,−21,−21) form an orthonormal basis for the subspace of R3 spanned by the vectors (1,−1,1) and (2,0,4).
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