32AH HW2 #2

My hope is that this picture will help with question 2 on the second homework:

The picture indicates the situation in $\mathbf{R}^3$, but the idea generalizes to $\mathbf{R}^n$ for any $n$. The idea is that the set $W$ is the $(n-1)$-dimensional hyperplane containing $x_0$ that is perpendicular to the vector $u$. The point $w_0$ is the point in $W$ closest to $x \in \mathbf{R}^n$.

Intuitively, we should expect that $w_0$ is the intersection of $W$ with the line containing $x$ perpendicular to $W$ (i.e. parallel to $u$). Your job is to formalize this intuition in terms of the vector geometry we’ve learned. As the hint in the problem suggests, a good strategy is to apply the Pythagorean theorem to the triangle $x\ w_0\ w$. Notice that in order to apply the Pythagorean theorem you must prove that the vectors $x – w_0$ and $w – w_0$ are orthogonal. Once you define $w_0$ (in terms of $x$, $x_0$ and $u$) you will also need to prove that $w_0 \in W$.

Will Rosenbaum

Saarbrücken, Germany

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