Let $k$ and $n$ be integers satisfying $0<k<n$, and let $P, Q \subseteq \mathbb{R}^{n}$ be the linear subspaces spanned by $\left(e_{1}, \ldots, e_{k}\right)$ and $\left(e_{k+1}, \ldots, e_{n}\right)$, respectively, where $e_{i}$ is the $i$ th standard basis vector for $\mathbb{R}^{n}$. For any $k$-dimensional subspace $S \subseteq \mathbb{R}^{n}$ that has trivial intersection with $Q$, show that the coordinate representation $\varphi(S)$ constructed in Example 1.36 is the unique $(n-k) \times k$ matrix $B$ such that $S$ is spanned by the columns of the matrix $\left(\begin{array}{c}I_{k} \\ B\end{array}\right)$, where $I_{k}$ denotes the $k \times k$ identity matrix.

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Problem 1.10

Let $k$ and $n$ be integers satisfying $0 < k < n$ , and let $P, Q \subseteq ℝ^{n}$ be the linear subspaces spanned by $(e_{1}, \dots, e_{k})$ and $(e_{k + 1}, \dots, e_{n})$ , respectively, where $e_{i}$ is the $i$ th standard basis vector for $ℝ^{n}$ . For any $k$ -dimensional subspace $S \subseteq ℝ^{n}$ that has trivial intersection with $Q$ , show that the coordinate representation $φ (S)$ constructed in Example 1.36 is the unique $(n - k) \times k$ matrix $B$ such that $S$ is spanned by the columns of the matrix $(\begin{matrix} I_{k} \\ B \end{matrix})$ , where $I_{k}$ denotes the $k \times k$ identity matrix.

Problem 1.10

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