So I understand when two subspaces are considered perpendicular and what it means for vectors to be perpendicular/orthogonal.
The question I have is, if two vectors are perpendicular, do they always have to exist in orthogonal subspaces such as the nullspace and rowspace (I am using nullspace and rowspace as examples)? Can orthogonal vectors exist in the same subspace?
Finally, if $A^T = A$, then is the column space $\perp$ to nullspace and left nullspace?
Given any two vectors $u, w$, you can for example consider $\text{span}(\{u,w \})$, which is a subspace that contains both $u$ and $w$. To answer your question more directly, take for example any two nonzero orthogonal vectors in $\mathbb{R}^3$. Then their span is a plane containing both vectors, but neither vector is contained in the line orthogonal to this plane.
For real matrices $A$, the row space is orthogonal to the null space. So if $A^T = A$, then yes, the column space is orthogonal to both the null space and left null space.
