Orthogonalizing a basis by gram schmidt process

[Kamekui]

Homework Statement

(a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}.
(b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations.

Homework Equations

gram schmidt orthogonalization and then normalizing

The Attempt at a Solution

(a.) I used gram schmidt orthogonalization and then normalized to get:

1/2{1,1,1,1}, 1/2{1,-1,-1,1}, 1/2{-1,1,-1,1}

(b.) I'm not sure how to do this, any help would be appreciated.

 

[LCKurtz]

Homework Statement

(a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}.
(b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations.

Homework Equations

gram schmidt orthogonalization and then normalizing

The Attempt at a Solution

(a.) I used gram schmidt orthogonalization and then normalized to get:

1/2{1,1,1,1}, 1/2{1,-1,-1,1}, 1/2{-1,1,-1,1}

(b.) I'm not sure how to do this, any help would be appreciated.

So you want$$
(2,2,2,2) = \frac {c_1}2 (1,1,1,1)+\frac {c_2}2 (1,-1,-1,1)+\frac {c_3}2 (-1,1,-1,1)$$What happens if you take the inner product of both sides of that with one of your basis vectors?