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Real Subspace Representations
Affine Set Dimensions
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Projections On Affine Sets
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Real Affine Set Representations


Since every affine set is a translate of a unique subspace, we can represent them by representing the vector and the subspace.


Recall that $M$ is affine means $M = S + a$ for some subspace $S$ and vector $a \in \R ^n$. The dimension of $M$ is the dimension of the subspace. Suppose $\dim S = k$, then there exists $Q \in \R ^{n \times k}$ with $Q^\top Q = I$, so that for any $x \in S$, there exists unique $z \in \R ^k$ with $x = Qz$. Since $M = S + a$, we have

\[ M = \Set{y \in \R ^n}{(\exists z \in \R ^k)(y = a + Qz)} \]

We also denote this set $\Set{a + Qz}{z \in \R ^k}$.

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