1 Prove that if x is orthogonal to each of vi v2 Vp then x

1. Prove that if x is orthogonal to each of vi, v2,... , Vp, then x is orthogonal to any vector in Span(vi, v2,.Vph

Solution

A.

x.v_i=0 for all i=1,...,p

Any vector in span{v1,...,vp} is, z=\\sum_{i=1}^p a_iv_i

x.z=\\sum_{i=1}^p a_i x.v_i=0

B.

1. 0 belongs to W^perpendicular

2. Let x, y belong to W^perpendicular

and w be any vector in W

(x+y).w=x.w+y.w=0+0=0

So, x+y is also in W^perpendicular

3.

Let x be in W^perpendicular and c be a real number.

For any w in W

(cx).w=c(x.w)=c*0=0

So, cx is also in W^perpendicular

Hence, W^perpendicular is a subspace

3)

Let x be in W intersection W perpendicular

So, x.x=xx^T=0

Let, x=(x1,...,xp)

xx^T=x1^2+....+xp^2=0

Since, xi are real for all i=1,..,p

Hence, xi=0 for all i=1,,...,p

Or x=0