I'm studying from my textbook, and from what I understand a hyperplane is a set of vectors with dimension $n-1$ that is the orthogonal complement of a set of vectors with dimension $n$. However there is this bit in the textbook that is really confusing me:
Since hyperplanes through the origin of $\Bbb{R}^n$ are the subspaces of dimension $n – 1$, their orthogonal complements are the subspaces of dimension 1, which are lines through the origin of $\Bbb{R}^n$.
I'm confused on one aspect. Shouldn't the dimension of the orthogonal complement of a hyperplane be $n$? Why does it say the orthogonal complement of a hyperplane are of dimension $1$, when it says earlier that the dimension of a hyperplane is $n-1$? I must not be understanding something correctly.
Best Answer
Any $k$-dimensional linear subspace of an $n$-dimensional vector space intersects its orthogonal complement trivially. This is because the only vector that is perpendicular to itself is the $0$-vector. Hence the orthogonal complement is at most $(n-k)$-dimensional. In particular this means the orthogonal complement of a hyperplane is at most $1$-dimensional, and certainly not $n$-dimensional.
That the orthogonal complement is in fact precisely $(n-k)$-dimensional can be seen from the fact that any orthogonal basis for the $k$-dimensional subspace can be extended to an orthogonal basis for the $n$-dimensional vector space itself, by the basis extension theorem and Gram-Schmidt.