Vectors that form bases (linear algebra)

[Niles]

[SOLVED] Vectors that form bases (linear algebra)

Homework Statement

I am given two vectors u and v, which are:

u = (1/2 , 1/2 , 1/2 , 1/2) and
v = (1/2 , 1/2 , -1/2 , -1/2).

I have to find an orthonormal basis for R^4 containing u and v.

The Attempt at a Solution

The first thing that came to me was Gram-Schmidt - but then I saw that the dot-product between u and v is zero, so Gram-Schmidt is overkill.

I just need to find two other linearly independent vectors that has dot-product equal zero with respectively u and v. Is that even possible and how would I do that?

 

[HallsofIvy]

Okay, u and v are already orthogonal so you really just need to find two more vectors that are orthogonal to both u and v (and to each other). Let w= (a, b, c, d). Then $u\cdot w$ is (1/2)a+ (1/2)b+ (1/2)c+ (1/2)d and you want that equal to 0. That, of course, is the same as a+ b+ c+ d= 0. $v\cdot w$ is (1/2)a+ (1/2)b- (1/2)c- (1/2)d= 0 which is the same as a+ b- c- d= 0. Adding the two equation, you get 2a+ 2b= 0 or b= -a. With b= -a, the first equation becomes c+ d= 0 or c= -d. Choose a and d to be whatever you want and you will get a vector orthogonal to both u and v. You should be able to choose one value of a and d to get a vector orthogonal to u and v, then choose another value of a and d to get a vector that is orthogonal to u and v and also to your first vector.