Linear Algebra: orthogonal components of a vector

[cesaruelas]

Homework Statement

Let V = Gen{ [0;5;1;2], [4;0;-2;1], [5,1,0,1]}. Define u1₁=v₁. Indicate the coordinates of u₂, the orthogonal component of v₂ to V₁=Gen{u₁.

Homework Equations

I know V has to be a vector space. If there is a subspace W with an orthogonal basis B={v₁,...,v_k} then the orthogonal component of any vector u to W is

u_c=u - (<u,v₁>/<v₁,v₁>)*v₁ - ... - (<u,v_k>/<v_k,v_k>)*v_k

The Attempt at a Solution

I don-t even know where to start since the basis given is not orthogonal. Do I have to construct an orthogonal basis from the three vectors given and then proceed to solve it? I didn't see this topic in class so I didn't see the professor do this, I'm doing this for extra credit on the subject. Anyone knows where I could find a worked out example of this type of problems? Thank you.

 

[HallsofIvy]

Since the problem is that the basis is not orthogonal, use the "Gram-Schmidt orthogonalization procedure".