###### Example 3.2.7

Suppose we work for a company that records its quarterly revenue, in millions of dollars, as:

Quarter | Revenue |

1 | 10.3 |

2 | 13.1 |

3 | 7.5 |

4 | 8.2 |

Rather than using a table to record the data, we could display it in a graph or write it as a vector in \(\real^4\text{:}\)

Let's now consider a new basis \(\bcal\) for \(\real^4\) using vectors

We may view these basis elements graphically, as in FigureĀ 3.2.9

As we wish to convert our revenue vectors into the coordinates given by \(\bcal\text{,}\) we form the matrices:

and compute

This means that our revenue vector is

We will think about what these coordinates mean by adding the basis vectors together one at a time.

The first coordinate gives us the average revenue over the year: \(9.775\vvec_1\text{.}\)

Adding in the second component shows how the averages in the first and second halves of year differ from the annual average: \(9.775\vvec_1 + 1.925\vvec_2\text{.}\)

The third and fourth components break down the behavior in the first and second halves of the year into quarters:

If we write \(\coords{\xvec}{\bcal} = \fourvec{c_1}{c_2}{c_3}{c_4}\text{,}\) we see that the coefficient \(c_1\) measures the average revenue over the year, \(c_2\) measures the deviation from the annual average in the first and second halves of the year, and \(c_3\) measures how the revenue in the first and second quarter differs from the average in the first half of the year. In this way, the coefficients provide a view of the revenue over different time scales, from an annual summary to a finer view of quarterly behavior.

This basis is sometimes called a *Haar* wavelet basis, and the change of basis is known as a *Haar* wavelet transform. In the next section, we will see how this basis provides a useful way to store digital images.