Find Basis of R^n Subspace with Sum X^2=1

[zairizain]

Homework Statement

Let V=(x1,x2,..Xn) Sum X = 0 be a subspace of R^n. Find a basis of V such that Sum X^2=1

Homework Equations

The Attempt at a Solution

for Sum X =0

x1+x2+..+xn =0

xn= -x1-x2-...

So <x1, x2, ...,-x1-x2-...>=<x1, 0,0...,-x1> +<0,x2,0,...,-x2)+...

=x1<1,0,0,...,-1>+x2<0,1,0,...-1>+...
norm, ||x1||=||x2||=||xn||=... = 2^(1/2)

Basis {<1/norm x1, 0, 0,..-1/norm x1>, <0,1/normx2,0,...-1/normx2>,...}For Sum X^2=0

x1^2 +x2^2+...+xn^2 =1
xn^2 = 1- (x1^2 +x2^2+...)

That all I can do. Please guide me.

 

[mighty2000]

Your solution for Sum X=0 looks correct. For Sum X^2=1, we can use the same approach as before. Let's start by writing out the equation:

x1^2 +x2^2+...+xn^2 =1

We can rewrite this as:

(x1^2-1)+(x2^2-1)+...+(xn^2-1)=0

Now, we can use our previous basis for Sum X=0 and simply add 1 to each element. This will give us a new basis where the sum of squares is equal to 1:

Basis {<1/norm x1 +1, 0, 0,..,-1/norm x1 +1>, <0,1/normx2 +1,0,...,-1/normx2 +1>,...}

Alternatively, we can use the Gram-Schmidt process to find an orthogonal basis for V. Starting with our original basis for Sum X=0, we can use the following algorithm:

1. Let v1=<1/norm x1, 0, 0,..,-1/norm x1>
2. Let u2=<0,1/normx2,0,...,-1/normx2>
3. Let v2=u2-<v1,u2>/<v1,v1>*v1
4. Normalize v2: v2=v2/||v2||
5. Continue this process for all remaining vectors in the original basis.

This will give us an orthogonal basis for V, and we can then normalize each vector to make it an orthonormal basis.