Today I feel like giving away a little quiz material on applied linear algebra. The topic is relevant these days wherein some misleading SEOs are playing the we-do-“science” game (quack “science”, after all).

The following is taken from the Search Engines Architecture grad course I lectured back in 2008. I’m providing only one exercise with multiple parts. The quiz with answers might be a great topic for an IRW issue.

1.1 A search engine has three types of revenue channels: pay-per-click (PPC), pay-per-placement (PPP), and pay-for-conversion (PFC). In quarter 1, the million-dollar revenues respectively were: 20, 4, and 9. In quarter 2, PPC revenues were 20% less, PPP revenues doubled, and PFC revenues remained constant.

1.1.1 Write a matrix M1 expressing the revenue and quarter vectors for the first two quarters.

1.1.2 If the goal in quarter 3 is to increase by 20% all revenues earned in quarter 2, update M1 so it reflects such a goal as a new matrix M2.

1.1.3 If the goal in quarter 4 is to meet the average revenues of each of the previous channels in quarter 4, update M2 such that it reflects that goal as a new matrix M3.

1.1.4 Express the above quarters as column unit vectors. Inspecting either rows or columns, construct a nearest neighbor similarity matrix Mnn and construct scalar clusters of quarters. Ignore cosine similarity deviations of 0.02 units or less. How similar the quarters are?

Have fun.