1. ## perpendicular subspaces

Im lost on this problem:
Let B be a basis for a subspace W of an inner product space V, and let z exist in V. Prove that z exists in W^perp if and only if <z,v>=0 for all v in B.
All I have is that <z,x>=0 for all x in W.
cant seem to figure out how to prove either direction of the statement though.

2. Originally Posted by dannyboycurtis
Im lost on this problem:
Let B be a basis for a subspace W of an inner product space V, and let z exist in V. Prove that z exists in W^perp if and only if <z,v>=0 for all v in B.
All I have is that <z,x>=0 for all x in W.
cant seem to figure out how to prove either direction of the statement though.

Write an element of W as a lin. combination of elements in B, and use linearity of the inner product.

Tonio

3. so I have that
x exists in W, where x=a_1v_1+...+a_nv_n for a's existing in the field V is defined on and v's vectors in the basis B.
then <z,x>=0 implies
<z,a_1v_1+...+a_nv_n>
=<z,a_1v_1>+...+<z,a_nv_n>
I cant see where to go after this, if I take out the scalar values, Im not sure where to proceed...

4. Originally Posted by dannyboycurtis
so I have that
x exists in W, where x=a_1v_1+...+a_nv_n for a's existing in the field V is defined on and v's vectors in the basis B.
then <z,x>=0 implies
<z,a_1v_1+...+a_nv_n>
=<z,a_1v_1>+...+<z,a_nv_n>
I cant see where to go after this, if I take out the scalar values, Im not sure where to proceed...

Yes, take the scalars out and in every summand you get $ = 0$, because this is precisely what you're assuming!

Tonio