a) Let W be a subspace of R^n and v a vector in R^n. Suppose that w and w\' are orthogonal
vectors with w in W and that v=w+w\'. Is it necessarily true that w\' is in W(orthogonal
complement)?
Either prove that it is true or find a counterexample.
b) Let {v1,....,vn} be an othogonal basis for R^n and let W=span(v1,...,vk). Is it necessarily true
that W(orthogonal complement) = span(v k+1,......,vn)?
Either prove that it is true or find a counterexample.
Solution
a) No, not necessary.
Take n = 3, and let W be the subspace of all points (x,y,z) in R3 with z = 0. Let v = (1,1,0) and
w = (1,0,0) and w\' = (0,1,0). Clearly, v = w + w\', and also (w,w\') = 0 so w,w\' are orthogonal.
But clearly w\' also is in W and w\' is not zero, hence it is not in Wperp.
b) Yes.
Since v1,v2,...vn form an orthogonal basis for Rn we have (vi,vj) = 0 whenever i and j are
distinct.
We know that dim(W) + dim(Wperp) = n. Since W is spanned by v1,v2,...,vk it follows that W
has dim = k
which implies that Wperp has dim = n-k. Moreover, since the vi are orthogonal, each vi is
orthogonal to vj where 0< i < k+1, k.
a) Let W be a subspace of R^n and v a vector in R^n. Suppose that w .pdf
1. a) Let W be a subspace of R^n and v a vector in R^n. Suppose that w and w' are orthogonal
vectors with w in W and that v=w+w'. Is it necessarily true that w' is in W(orthogonal
complement)?
Either prove that it is true or find a counterexample.
b) Let {v1,....,vn} be an othogonal basis for R^n and let W=span(v1,...,vk). Is it necessarily true
that W(orthogonal complement) = span(v k+1,......,vn)?
Either prove that it is true or find a counterexample.
Solution
a) No, not necessary.
Take n = 3, and let W be the subspace of all points (x,y,z) in R3 with z = 0. Let v = (1,1,0) and
w = (1,0,0) and w' = (0,1,0). Clearly, v = w + w', and also (w,w') = 0 so w,w' are orthogonal.
But clearly w' also is in W and w' is not zero, hence it is not in Wperp.
b) Yes.
Since v1,v2,...vn form an orthogonal basis for Rn we have (vi,vj) = 0 whenever i and j are
distinct.
We know that dim(W) + dim(Wperp) = n. Since W is spanned by v1,v2,...,vk it follows that W
has dim = k
which implies that Wperp has dim = n-k. Moreover, since the vi are orthogonal, each vi is
orthogonal to vj where 0< i < k+1, k
