Orthogonal Spaces and Complements

Orthogonal subspaces

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


The xx-axis and yy-axis in R2\R^2 are orthogonal subspaces. In particular, let v=[α0]\bold v=\begin{bmatrix}\alpha\\0\end{bmatrix} and u=[0β]\bold u=\begin{bmatrix}0\\\beta\end{bmatrix} be any vectors along the xx-axis and yy-axis, respectively. We can see that the dot product of u\bold{u} and v\bold{v} is 00, implying that u\bold{u} and v\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 R3\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.

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