Find orthogonal base for u+v without answer

[transgalactic]

i got these vectors R4:
v1=(1,1,0,1)
v2=(0,-4,2,0)
v3=(0,0,-18,0)
u1=(1,1,0,1)
u2=(-1,0,1,0)


in the row reduction process v1 v2 v3 u1 are left independent
i need to find the orthogonal base of u+v?

in the solution i was told that because v1 v2 v3 u1 are independent

then the orthogonal base of u+v the standard base of is
(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

but why take the standart base
the vectors v1 v2 v3 u1 are independent they can act as a base
??

 

[HallsofIvy]

You have left some things out. Are we to assume that v is spanned by v1, v2, and v3? And that u is spanned by u1 and u2?

Yes, v1, v2, v3, u1 are independent and so are a basis for u+v. But they are not orthogonal. Since there are four of them, they span all of R⁴. You know that the standard basis for R⁴ is orthonormal so it satisfies the conditions of the problem: Find an othogonal basis for u+ v.