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One familiar concept from geometry is that of flat spaces such as lines and planes. By analogy,
we can consider higher-dimensional analogues, as well as the 0-dimensional equivalent, which is easily
shown to be a point.1
An n-dimensional space of this type is symbolized as Rn
; since all such spaces are
identical for a given n, it is treated as a single object rather than a category. Extending the analogy, just
as we can consider a line drawn through the origin of a 2-dimensional graph, we can consider a copy of
R
k
going through the origin of Rn
(for k<n). We can similarly consider the set of all such copies of Rk
for a
given k and n. Such a copy of Rk
going through the origin of Rn
is known as a linear subspace of Rn
, and
the set of all such linear subspaces is known as a Grassmanian and denoted Gr(n,k). These linear
subspaces can in turn have lower-dimensional subspaces of their own; the resultant structure shows
certain similarities to the structure of subsets of a given set .

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