and neither restricts to the Grassmann distance when the subspaces are of the same dimension. Our main objective is to show that there is an alternative definition that does generalize the Grassmann distance but our work will also shed light on these two distances.

Main Contributions
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Our main result (see Theorem \[thm1\]) can be stated in simple linear algebraic terms: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$, assuming $k< l$ without loss of generality, the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives the distance between $\mathbf{A}$ and $\mathbf{B}$. Taking an algebraic geometric point-of-view:

1.  \[quote\] *The distance between subspaces of different dimensions is the distance between a point and a certain Schubert variety within the Grassmannian.*

This distance has the following properties, established in Section \[sec:main\]:

1.  readily computable via <span style="font-variant:small-caps;">svd</span>;

2.  restricts to the usual Grassmann distance for subspaces of the same dimension;

3.  independent of the choice of local coordinates;

4.  independent of the dimension of the ambient space (i.e., $n$);

5.  \[other\] may be defined in 