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Real Matrices with Orthonormal Columns
Needed by:
Projections On Subspaces
Real Affine Set Representations
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Real Subspace Representations


How should we represent a subspace computationally?


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

\[ Q = \bmat{q_1 & q_2 & \cdots & q_k}. \]

For every $x \in S$, there exists unique coefficients $z \in \R ^k$ so that

\[ x = Qz. \]

Therefore we have a one-to-one correspondence between vectors $x \in S$ and their coordinates $z \in \R ^k$.

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