Question

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?

Answer 1

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

Answer 2

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

Answer 3

Oh my prof skipped that chapter hahaha

Answer 4

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!

Answer 5

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.

Answer 6

and that will give us u2* correction :))