Orthogonal Spaces and Complements

Orthogonal subspaces

Two subspaces, $U$ and $V$, of a vector space are orthogonal when every vector in $U$ is orthogonal to every vector in $V$ and the other way around.

Example

The $x$-axis and $y$-axis in $\R^2$ are orthogonal subspaces. In particular, let $\bold v=\begin{bmatrix}\alpha\\0\end{bmatrix}$ and $\bold u=\begin{bmatrix}0\\\beta\end{bmatrix}$ be any vectors along the $x$-axis and $y$-axis, respectively. We can see that the dot product of $\bold{u}$ and $\bold{v}$ is $0$, implying that $\bold{u}$ and $\bold{v}$ are orthogonal.

Note: It’s sufficient to test the orthogonality of the basis of two subspaces to establish the orthogonality of the subspaces.

Planes at the right angle

We may make a mistake if we visualize two planes in $\R^3$ that look orthogonal, as shown in the visualization below. However, these subspaces aren’t orthogonal, because a line of intersection is common to both the planes. The vectors along the line of intersection break the orthogonality between the subspaces.