  This note proves that, for $F = \Bbb{R,C}$ or $\Bbb{H}$, the bordism classes
of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k < n$ and
having real dimension $d$, constitute a linearly independent set in the
unoriented bordism group ${\frak{N}}_d$ regarded as a ${\Bbb{Z}}_2$-vector
space.
