Solving Problems with Orthogonal Vectors

[gbacsf]

I need some help in understanding what I need to do to solve these poblems, I can't get them started.

1. Find an orthogonal set of vectos that spans the same subspace as a,b,c.

a=(1,1,-1)
b=(-2,-3,1)
c=(-1,-2,0)

2. Use the Gram-Schmidt process to find and orthogonal basis that contains the vecto (-1,-4,2,-4)

 

[0rthodontist]

In 1., this IS the gram-schmidt process. Just apply it.

In 2., you have to pick some basis that contains that vector (try standard normal except for the vector) and then apply the gram-schmidt process.

 

[gbacsf]

Thanks,

I finished off #1.

For the second question, what is meant by standard normal? Does that mean to use (0,1,0,0), (0,0,1,0) and (0,0,0,1) with the original vector in gram-schmidt to find the orthogonal basis?

 

[HallsofIvy]

Yes. Or (1, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) and the given vector, etc. In other words, any basis containing the given vector. By choosing 3 vectors from the "standard" basis you know they are independent, by making sure that your given vector has a non-zero number where the "missing" vector has a 1 you know that all 4 are independent and therefore, a basis.