How do I show $QQ^T$ as an orthogonal projector? Do I take the dot product? I'm not sure how to answer this question
Suppose that $Q$ is an $n$ x $k$ matrix whose columns form an orthonormal basis for a subspace $S$ of $R^n$. Show that $QQ^T$ is an orthogonal projector onto $S$
As should be known to anyone faced with such an exercise, the matrix $P=QQ^T$ represents an orthogonal projection if and only if $P^2=P$ and $P$ is self-adjoint. It is easy to check that $P$ is self-adjoint because $$P^T=(QQ^T)^T=(Q^T)^TQ^T=QQ^T=P$$ What about $P^2$? $$P^2=QQ^TQQ^T$$ What can we say about the middle part, $Q^TQ$? We can say that it is the $k\times k$ identity matrix, because the columns of $Q$ are orthonormal. It follows that $$P^2=Q\times I_k\times Q^T=QQ^T=P$$ and we are done.