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## Homework Statement

Find an orthonormal basis for the subspace of V

_{4}spanned by the given vectors.

x

_{1}= (1, 1, 0, 1)

x

_{2}= (1, 0, 2, 1)

x

_{3}= (1, 2, -2, 1)

## Homework Equations

Gram-Schmidt Process

## The Attempt at a Solution

I have used the Gram-Schmidt process but seem to be running into trouble. Here is what I did:

y

_{1}= x

_{1}= (1, 1, 0, 1)

y

_{2}= x - y

_{1}= (1-1, 0-1, 2-0, 1-1) = (0, -1, 2, 0)

y

_{3}= x

_{3}- y

_{1}+ y

_{2}= (1-1+0, 2-1-1, -2-0+2, 1-1+0) = (0, 0, 0, 0)

Now I used these and their norms to find the basis {y

_{1}, y

_{2}}

y

_{1}/ lly

_{1}ll = 1/sqrt(3) (1, 1, 0, 1)

y

_{2}/ lly

_{2}ll = 1/sqrt(5) (0, -1, 2, 0)

Therefore, {1/sqrt(3) (1, 1, 0, 1), 1/sqrt(5) (0, -1, 2, 0)} from my work. However, my book says the answer is {(1/3)(1/sqrt(3)(1, 1, 0, 1), 1/sqrt(42) (1, -2, 6, 1)} which is very different than my answer. Where am I going wrong?