Suppose $Q_{1}$ is an $n$ x $p $ matrix (derived from the QR Decomposition of X) whose columns provide an orthonormal basis for the subspace ${\chi}$ of $R^{n}$ spanned by the columns of an $n$ x $p$ matrix $X$ = $(x_1,...,x_p)$. The hat matrix $H$ = $Q_{1}Q_{1}^{T}$ projects vectors orthogonally onto $X$.

Suppose the first two rows of $X$ are the same. Explain why the first two rows of $H$ are the same.

Suppose $Q_{1}$ is an $n$ x $p $ matrix (derived from the QR Decomposition of X) whose columns provide an orthonormal basis for the subspace ${\chi}$ of $\mathbb{R}^{n}$ spanned by the columns of an $n$ x $p$ matrix $X$ = $(x_1,...,x_p)$. The hat matrix $H$ = $Q_{1}Q_{1}^{T}$ projects vectors orthogonally onto $X$.

Suppose the first two rows of $X$ are the same. Explain why the first two rows of $H$ are the same.