$X$ is $n \times (k+1)$, and $X^TX$ is invertible. So what's the rank of $ I - X(X^TX)^{-1}X^T$, which $I$ is diagonal standard matrix.
2026-03-27 03:44:28.1774583068
what's the rank of $ I - X(X^TX)^{-1}X^T$
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Hint: If $P$ is the matrix of the orthogonal projection onto a subspace, then $I - P$ is the matrix of the projection onto the orthogonal complement of this subspace.