Linear Algebra, 2020a

300 Chapter Three. Maps Between Spaces
8 This table lists the average distance from the sun to each of the first seven planets,
using Earth’s average as a unit.
Mercury Venus Earth Mars Jupiter Saturn Uranus
0.39 0.72 1.00 1.52 5.20 9.54 19.2
(a) Plot the number of the planet (Mercury is 1, etc.) versus the distance. Note
that it does not look like a line, and so finding the line of best fit is not fruitful.
(b) It does, however look like an exponential curve. Therefore, plot the number
of the planet versus the logarithm of the distance. Does this look like a line?
(c) The asteroid belt between Mars and Jupiter is what is left of a planet that
broke apart. Renumber so that Jupiter is 6, Saturn is 7, and Uranus is 8, and
plot against the log again. Does this look better?
(d) Use least squares on that data to predict the location of Neptune.
(e) Repeat to predict where Pluto is.
(f) Is the formula accurate for Neptune and Pluto?
This method was used to help discover Neptune (although the second item is
misleading about the history; actually, the discovery of Neptune in position 9
prompted people to look for the “missing planet” in position 5). See [Gardner, 1970]
9 Suppose that W is a subspace of R n for some n and suppose that ⃗v is not an
element of W. Let the orthogonal projection of ⃗v into W be the vector proj W (⃗v) =⃗p.
Show that ⃗p is the element of W that is closest to ⃗v.