The American Heritage Dictionary of the English Language (Third Edition) gives two definitions of the word “archetype”: 1. An original model or type after which other similar things are patterned; a prototype; and 2. An ideal example of a type; quintessence.
Either use might apply here. Our archetypes are typical examples of systems of equations, matrices and linear transformations. They have been designed to demonstrate the range of possibilities, allowing you to compare and contrast them. Several are of a size and complexity that is usually not presented in a textbook, but should do a better job of being “typical.”
We have made frequent reference to many of these throughout the text, such as the frequent comparisons between Archetype A and Archetype B. Some we have left for you to investigate, such as Archetype J, which parallels Archetype I.
How should you use the archetypes? First, consult the description of each one as it is mentioned in the text. See how other facts about the example might illuminate whatever property or construction is being described in the example. Second, each property has a short description that usually includes references to the relevant theorems. Perform the computations and understand the connections to the listed theorems. Third, each property has a small checkbox in front of it. Use the archetypes like a workbook and chart your progress by “checking-off” those properties that you understand.
The next page has a chart that summarizes some (but not all) of the properties described for each archetype. Notice that while there are several types of objects, there are fundamental connections between them. That some lines of the table do double-duty is meant to convey some of these connections. Consult this table when you wish to quickly find an example of a certain phenomenon.
| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | |
| Type | S | S | S | S | S | S | S | S | S | S | M | M | L | L | L | L | L | L | L | L | L | L | L | L |
| Vars, Cols, Domain | 3 | 3 | 4 | 4 | 4 | 4 | 2 | 2 | 7 | 9 | 5 | 5 | 5 | 5 | 3 | 3 | 5 | 5 | 3 | 5 | 6 | 4 | 3 | 4 |
| Eqns, Rows, CoDom | 3 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 4 | 6 | 5 | 5 | 3 | 3 | 5 | 5 | 5 | 5 | 4 | 6 | 4 | 4 | 3 | 4 |
| Solution Set | I | U | I | I | N | U | U | N | I | I | ||||||||||||||
| Rank | 2 | 3 | 3 | 2 | 2 | 4 | 2 | 2 | 3 | 4 | 5 | 3 | 2 | 3 | 2 | 3 | 4 | 5 | 2 | 5 | 4 | 4 | 3 | 3 |
| Nullity | 1 | 0 | 1 | 2 | 2 | 0 | 0 | 0 | 4 | 5 | 0 | 2 | 3 | 2 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 0 | 1 |
| Injective | X | X | N | Y | N | Y | N | Y | X | Y | Y | N | ||||||||||||
| Surjective | N | Y | X | X | N | Y | X | X | Y | Y | Y | N | ||||||||||||
| Full Rank | N | Y | Y | N | N | Y | Y | Y | N | N | Y | N | ||||||||||||
| Nonsingular | N | Y | Y | Y | N | |||||||||||||||||||
| Invertible | N | Y | Y | Y | N | N | Y | Y | Y | N | ||||||||||||||
| Determinant | 0 | -2 | -18 | 16 | 0 | -2 | -3 | 0 | ||||||||||||||||
| Diagonalizable | N | Y | Y | Y | Y | Y | Y |