Exercise 5

Suppose that you have the data points $$(2,0), (2,3), (4,1), (3,2), (4,4).$$

1. Find the demeaned data points.

2. Find the total variance $V$ of the dataset.

3. Find the variance in the direction $\evec_1 = \twovec10$ and the variance in the direction $\evec_2=\twovec01$.

4. Project the demeaned data points onto the line defined by $\vvec_1=\twovec21$ and find the variance of these projected points.

5. Project the demeaned data points onto the line defined by $\vvec_2=\twovec1{-2}$ and find the variance of these projected points.

6. How and why are the results of from the last two parts related to the total variance?