Real Subspace Representations
Why
How should we represent a subspace computationally?
Definition
Given a subspace $S \in \R ^n$, since it is
finite dimensional, there exists a finite basis
for the space.
This basis can be made orthonormal.
Therefore every subspace $S$ has an orthonormal
basis $q_1, \dots , q_k$, where $k$ is the
dimension of the subspace.
We can stack these as a matrix. Define $Q \in
\R ^{n \times k}$ by
\[
Q = \bmat{q_1 & q_2 & \cdots & q_k}.
\] \[
x = Qz.
\]