I have this problem, where the author asks in two subspaces $W$ and $Z$, both having dimension 3, of a $5$ dimension space, is there always an element in $Z$ which is normal to the whole subspace $W$? My answer was yes, but the author considers the case, $W$=$Z$, my problem is that author mentioned them as different subspaces since he used different letters for those subspaces. Then how can he consider both as equal? is it allowed generally? is it because it was not mentioned specifically that the subspaces cannot be equal?
and is my answer correct, if subsapces have to be different?
Let's first look at the case of two different 2D subspaces, $W$ and $Z$, of a 3D space. The question is: If $W\ne Z$, is there always a vector $x \in Z$ such that $x \perp y$ for all $y\in W$ ?
The answer is no. Let $W=span(\{1,0,0\},\{0,1,0\})$ and $Z=span(\{1,0,0\},\{0,1,1\})$. We clearly have $W\ne Z$, but there is no vector is $Z$ that is orthogonal to the subspace $W$.
Expanding to higher dimensions, if $W$ and $Z$ are 3D subspaces of a 5D spaces, let $w_1$, $w_2$, $w_3$ be the orthogonal basis of $W$ and $z_1$, $z_2$, $z_3$ be the orthogonal basis of $Z$. That is , $$W=span(w_1,w_2,w_3) ,\quad Z=span(z_1,z_2,z_3)$$
If $w_1=z_1$ , $w_2=z_2$, and $w_3\not\perp z_3$,
you will have two different subspaces $W$ and $Z$ with no $x\in Z$ $| x\perp y$ $\forall$ $y\in W $