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Do the vectors v1 =   1 2 3   , v2 = ...

Do the vectors v1 =   1 2 3   ,

v2 =   √ 3 √ 3 √ 3   ,

v3   √ 3 √ 5 √ 7   ,

v4 =   1 0 0   form a basis for R 3 ? Why or why not?

(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and a2, where a1 =     1 0 −1 0     , a2 =     0 1 0 −1     . Find a basis for the orthogonal complement V ⊥ in R 4 .

(c) Find a basis for R 4 that includes the vectors a1, and a2. Your answer should be justified.

(d) Explain in words how to generalize your computation in part (c) to obtain a basis for R n that includes a given pair of orthogonal vectors u, v ∈ R n.

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