Linear Algebra, 2020a

Topic
Line of Best Fit
This Topic requires the formulas from the subsections on Orthogonal Projection
Into a Line and Projection Into a Subspace.
Scientists are often presented with a system that has no solution and they
must find an answer anyway. More precisely, they must find a best answer.
For instance, this is the result of flipping a penny, including some intermediate
numbers.
number of flips 30 60 90
number of heads 16 34 51
Because of the randomness in this experiment we expect that the ratio of heads
to flips will fluctuate around a penny’s long-term ratio of 50-50. So the system
for such an experiment likely has no solution, and that’s what happened here.
30m = 16
60m = 34
90m = 51
That is, the vector of data that we collected is not in the subspace where ideally
it would be.
⎛
⎜
16
⎞ ⎛
⎟ ⎜
30
⎞
⎟
⎝34⎠ ∉ {m ⎝60⎠ | m ∈ R}
51 90
However, we have to do something so we look for the m that most nearly works.
An orthogonal projection of the data vector into the line subspace gives a best
guess, the vector in the subspace closest to the data vector.
⎛
⎜
16
⎞ ⎛ ⎞
30
⎟ ⎜ ⎟
⎝34⎠ • ⎝60⎠
⎛ ⎞ ⎛
51 90
30
⎛
⎜
30
⎞ ⎛ ⎞
⎜ ⎟
· ⎝60⎠ = 7110
30
12600 · ⎜
30
⎞
⎟
⎝60⎠
⎟ ⎜ ⎟ 90
90
⎝60⎠ • ⎝60⎠
90 90