### Least Squares

#### Part 4: Summary

- Suppose you have n pairs
of data points,

(X_{1},Y_{1}), (X_{2},Y_{2}), ..., (X_{n},Y_{n}).
Describe in your own words the problem of finding the best fitting line
for this data. In what vector space can this be formulated as a linear
algebra problem? What linear algebra problem is equivalent to finding the
coefficients of the least squares line?

- How are the normal equations
constructed from the data? For fitting a line to data, how many normal
equations are there, and how many variables do they have?
- Suppose X
_{1} = X_{2} = ... = X_{n}. What will the matrix X look like? What will the data plot look like? Explain why you would expect the normal equations to be inconsistent in this situation.
- What geometric relationship
is expressed by the normal equations? Why do the normal equations almost
always have a unique solution?

modules at math.duke.edu