$k < l$, we show that the probability a random $l$-dimensional subspace contains $\mathbf{A}$ *equals* the probability a random $k$-dimensional subspace is contained in $\mathbf{B}$.

The far-reaching work [@EAS] popularized the basic *differential geometry* of Stiefel and Grassmannian manifolds by casting the discussions concretely in terms of matrices. Subsequent works, notably [@AMSbook; @AMS; @AMSV], have further enriched this concrete matrix-based approach. A secondary objective of our article is to do the same for the basic *algebraic geometry* of Grassmannians. In particular, we introduce some of the objects in Table \[tab:objects\] to an applied and computational mathematics readership. The proofs of our main results essentially use only the <span style="font-variant:small-caps;">svd</span>. Everything else is explained within the article and accessible to anyone willing to accept a small handful of unfamiliar terminologies and facts on faith.

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  *Grassmannian*                   $\operatorname{Gr}(k,n)$                      models $k$-dimensional subspaces in $\mathbb{R}^n$                                                            §\[sec:Grass\]
  *Infinite Grassmannian*          $\operatorname{Gr}(k,\infty)$                 models $k$-dimensional subspaces regardless of ambient space                                                  §\[sec:infty\]
  *Doubly-infinite Grassmannian*   $\operatorname{Gr}(\infty,\infty)$            models subspaces of all dimensions regardless of ambient space                                                §\[sec:dinfty\]
  *Flag variety*                   $\operatorname{Flag}(k_1,\dots,k_m,n)$        models nested sequences of subspaces in $\mathbb{R}^n$; $\operatorname{Flag}(k,n) = \operatorname{Gr}(k,n)$   §\[sec:schubert\]
  *Schubert variety*               $\Omega(\mathbf{X}_1,\dots,\mathbf{X}_m,n)$   ‘linearly constrained’ subset of $\operatorname{Gr}(k,n)$                                                     §\[sec:schubert\]
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Grassmannian of linear subspaces 