MATH 245 midterm post-mortem

The midterm had 6 questions: 3 proofs done in class (Cauchy-Schwarz, orthogonal projection theorem, and parallelotope volume theorem — these can all be found in my notes), a question about simplex centroids, a question about a “sum-of-squares”-minimizing point in the plane, and a question about angles between orthogonal complements (which no one claimed to have solved).

I’ll post the questions (#4 to #6, since as I said the rest were just proofs from class) here. Feel free to post your solutions, or any ideas. If I have time, I’ll post my solution to #4 below (although it took me a lot of algebra to prove uniqueness…)

  1. [#4 on exam] Let [a_0, \ldots, a_\ell] be an \ell-simplex with centroid g. For each k with 0 \leq k \leq \ell, define g_k to be the centroid of the (\ell-1)-simplex \zeta_k = [a_0, \ldots, a_{k-1}, a_{k+1}, \ldots, a_\ell].  Define also L_k to be the line passing through a_k and g_k. Show that the lines L_k all intersect uniquely at the point g.
  2. [#5 on exam] Given a_1, \ldots, a_\ell \in \mathbb{R}^2, find a point x \in \mathbb{R}^2 minimizing the sum \sum\limits_{i=1}^\ell |x-a_i|^2. Prove this point is unique.
  3. [#6 on exam] Let U, V be hyperspaces in \mathbb{R}^n. Prove that \mathsf{angle}(U,V) = \mathsf{angle}(U^\perp, V^\perp).

Also, sorry about the horribly outdated Spherical Geometry section in my notes. There are so many diagrams in my notebook that I haven’t had the time to transplant into LaTeX. I’ll do it the minute I get some spare time. Alternatively, you can just look at Professor New’s notes

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About mlbaker

just another guy trying to make the diagrams commute.
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One Response to MATH 245 midterm post-mortem

  1. dx7hymcxnpq says:

    First comment: my attempt at #5 was lame; I tried to use multivariable calc. Does anyone have a pure linear-algebra proof of #5?

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