## Romero 3 years ago Given a basis for the subpace H, if it's not an orthogonal set what can you do in order to make it an orthogonal set or is that not possible?

1. anonymoustwo44

It is possible :D do you know the gram-schimdt process?

2. LagrangeSon678

(depending) you could also simply multiplly each vector in the base by its magnitude

3. Romero

Oh my prof skipped that chapter hahaha

4. Romero

Ill learn it later. If she didnt teach it its probably not going to be in a test. You can post a useful link if you would like I'll read it later. Thanks!

5. anonymoustwo44

it goes like this, suppose we are given a basis B={v1,v2,..,vn}. Now to make this an orthogonal set of vectors which we will call O={u1,u2,..,un}, we do the following: let u1=v1 now let W={u1} so we know that component v2 orthogonal to W(hence orthogonal to all vectors in it)=v2-orthogonal projection of v2=v2-<v2,u1>u1/norm(u1)^2 and that will give us u1 so u2=v2-<v2,u1>u1/norm(u1)^2 for u3, let W={u1,u2} so the component of v3 orthogonal to W=v3-<v3,u1>u1/norm(u1)^2-<v3,u2>u2/norm(u2)^2 continue this process till vn now you can see the pattern already right? so to construct an orthogonal basis O={u1,u2,...,un} given a basis B={v1,v2,...,vn}, we do the following: u1=v1 u2=v2-<v2,u1>u1/norm(u1)^2 u3=v3-<v3,u1>u1/norm(u1)^2-<v3,u2>u2/norm(u2)^2 continue this process till we get to: un=vn-<vn,u1>u1/norm(u1)^2-<vn,u2>u2/norm(u2)^2-...-<vn,un-1>un-1/norm(un-1)^2 only the norm is squared. the <u,v>'s are just inner products.

6. anonymoustwo44

and that will give us u2* correction :))