### Least Squares

#### Part 2: Modeling the cancer
death data

In Part 1 of this module,
we saw that we could find the coefficients m and b of the least squares
line y = mx + b by finding the projection **p** of the data vector **y**
on the span W of the vectors **1** and **x** in R^{9}. In
fact, **p** = b**1** + m**x** -- that is, the coefficients are
also the coordinates of **p** with respect to the basis {**1**, **x**}
of W.

We saw in Part 3 of the
Orthogonality module that the projection **p** is UU^{T}**y**,
where U is a matrix whose columns are an orthonormal basis for W. In this
part we find p this way. Of course, our given basis {**1**, **x**}
is not orthonormal, so we have to start by constructing U.

- Enter the data vectors
**x** and **y**, as well as the vector **1** (called **one**
in your worksheet). Following the data entry step, you will find code for
constructing U. Explain the steps of this construction carefully.
- Find the projection
**p**.
The entries of this vector are related to something in the figure above,
which shows the original data points and the line y = mx + b. Explain carefully
where the entries of **p **are found in the figure.

modules at math.duke.edu