Exercise 1.8

Exercise 1.8. Let $0<k<n$ be integers, and let $P,Q\subset {ℝ}^n$ be the subspaces spanned by $(e_1,...,e_k)$ and $(e_{k+1},...,e_n)$, respectively, where $e_i$ is the ith standard basis vector. For any $k$-dimensional subspace $S\subset {ℝ}^n$ that has trivial intersection with $Q$, show that the coordinate representation $\mathrm{\Phi }(S)$ constructed in the preceding example (Grassmannian) is the unique $(\mathit{𝑛¯𝑘})\times k$ matrix $B$ such that $S$ is spanned by the columns of the matrix $\left(\begin{array}{c}\hfill I_k\hfill \\ \hfill B\hfill \end{array}\right)$, where $I_k$, denotes the $k\times k$ identity matrix.