Recent content by nlykkei

Ahh, yeah we have ker L is a subset of S := orthogonal complement of span(v) and dim(ker L) = dim(S)) implies ker L = S ? So u in S implies L(u) = 0, so for any w in R^n we have that L(w) is mapped to the orthogonal projection of w on span(v) ! And the matrix for this transformation is indeed...

dim(ker L) = n - 1, since dim(L(R^n)) = 1. But this doesn't show that L is given by v v^T for some vector v in L(R^n) ? In the notation from above, how can I use this to conclude L(j) = 0 ? Also I know ker L is a subset of the orthogonal complement to span(v) - But how do I show the opposite ?

Indeed it is. But how do I show that every vector in R^n is mapped equal by L and v v^T ? I know I can write a vector h = i + j, where i lies in Span(v) and j lies in the orthogonal complement of Span(v). Then L(h) = L(i) + L(j), but i = cv and suppose u is mapped to v: v = L(u) = L(L(u) =...

A linear operator L:Rn→Rn is called a projection if L^2=L. A projection L is an orthogonal projection if ker L is orthogonal to L(Rn). I've shown that the only invertible projection is the identity map I_Rn by using function composition on the identity L2(v)=L. Question: Now suppose that L...