Space Orthogonality
Two vectors (sub)spaces can also be orthogonal
Consider two subspaces $S$ and $T$
 $S \; \bot \; T$ means that $\forall \mathbf s \in S, \forall \mathbf t \in T: \mathbf s \; \bot \; \mathbf t$
 i.e. if every vector in $T$ is orthogonal to every vector in $S$, then $S$ and $T$ are orthogonal
Examples
Example 1
Suppose you have two spaces: a wall and a floor. Are they orthogonal?

 take one vector from the wall that is 45° to one of the axis. It's not orthogonal to the floor! it's 45°
 also, there are vectors that belong to both subspaces (and not just the origin!)  these vectors are not orthogonal
So, if two spaces intersect in more than just the zerovector, then they cannot be orthogonal
Example 2
Two subspaces that meet in $\mathbf 0$ can be orthogonal
Row space $C(A^T)$ and nullspace $N(A)$ are orthogonal.
why?
Let's consider only rows from $A$
 $\mathbf x \in N(A) \Rightarrow A \mathbf x = \mathbf 0$
 $\begin{bmatrix}
— (\text{row 1}) \,— \\
— (\text{row 2}) \,— \\
\vdots \\
— (\text{row $n$}) \,—
\end{bmatrix} \cdot \mathbf x = \begin{bmatrix}
0 \\ 0 \\ \vdots \\ 0
\end{bmatrix}$
(\text{row 1})^T \mathbf x = 0 \\
(\text{row 2})^T \mathbf x = 0 \\
\vdots \\
(\text{row $n$})^T \mathbf x = 0 \\
\end{bmatrix}$
 so $\mathbf x$ is orthogonal to all rows in $A$
what else is in the row space? linear combinations of rows of $A$
 $c_1 \cdot \text{row 1} + \ ... \ + c_n \cdot \text{row $n$}$ what if we multiply it by $\mathbf x$?
 $(c_1 \cdot \text{row 1} + \ ... \ + c_n \cdot \text{row $n$})^T \mathbf x = (c_1 \cdot \text{row 1})^T \mathbf x + \ ... \ + (c_n \cdot \text{row $n$})^T \mathbf x = c_1 \cdot \underbrace{(\text{row 1})^T \mathbf x}_{0} + \ ... \ + c_n \cdot \underbrace{(\text{row $n$})^T \mathbf x}_{0} = 0$
They are orthogonal for exactly the same reason
 just transpose $A$ and go through the same argument as for row space and nullspace
Orthogonal Compliments
$N(A) \; \bot \; C(A^T)$ and $\text{dim} N(A) + \text{dim} C(A^T) = n$
 they add up to the whole space
 so they are orthogonal compliments in $\mathbb R^n$
Sources