Sometimes, life is not linear. In this chapter, we'll model problems as linear processes, which will lead to linear systems such as Ax = b. Since our data won't be exactly linear, a solution won't always exist to such a system. When a solution doesn't exist, there does not exist a vector x such that Ax equals b. In other words, if we could find such an x then Ax – b = 0, which is the zero vector. Since we cannot achieve the zero vector by choosing x appropriately, our task will be to minimize the length of the vector Ax – b. We apply this technique first in a quest to approximate when the fastest 100 meter race might be run, and later we look at the rankings for the best colleges in the United States.

Dash of Math

In 2012, Usain Bolt electrified the Olympic track and field stadium in London as he won a second consecutive gold medal in the 100 meter dash. This was the fastest time to date ever in the Olympics. No previous medalist could have beat him.

There are 28 gold medal times for the men's 100 m race in the Olympic Games between 1896 and 2012 with the times listed in Table 7.1. The slowest time was Tom Burke's 12 second sprint to gold in 1896. Bolt was the fastest in 2012. Let's get a sense of all the times by graphing them as seen in Figure 7.1 (a). We see an overall trend of decreasing times. It's a trend, so it may not always be true. In 1968, Jim Hines won gold in 9.95, which was the first sub-10 second time. It wouldn't be until 1984 and Carl Lewis that another gold would be won with a race under 10 seconds.

There is another trend in the data. It can be approximated by a line. While the line won't pass through every point, it can get close.