Find an orthonormal basis for the vector subspace of $\displaystyle \mathbb R^3,\,\mathbb W=\{(x,y,z)\in\mathbb R^3|x+y+z=0\}.$

Sketch of work:

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- Jun 12th 2009, 05:30 PMMorganOrthonormal basis
Find an orthonormal basis for the vector subspace of $\displaystyle \mathbb R^3,\,\mathbb W=\{(x,y,z)\in\mathbb R^3|x+y+z=0\}.$

Sketch of work:

__Spoiler__: - Jun 12th 2009, 08:13 PMNonCommAlg
first, you don't need to hide your own work! second, you're right so far. now you just need to apply Gram-Schmidt process. let's give a name to the elements of the basis you found:

$\displaystyle u_1=(-1,1,0), \ u_2=(-1,0,1).$ now find $\displaystyle v_1=u_2-\frac{<u_1,u_2>}{<u_1,u_1>}u_1.$ the set $\displaystyle \{u_1,v_1 \}$ is an orthogonal basis for $\displaystyle W.$ so you just need to normalize them, i.e. find $\displaystyle e_1=\frac{u_1}{||u_1||}, \ e_2=\frac{v_1}{||v_1||}.$

then $\displaystyle \{e_1,e_2 \}$ would be an orthonormal basis for $\displaystyle W.$