Archetypes
WordNet (an open-source lexical database) gives the following definition of "archetype": something that serves as a model or a basis for making copies.
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.
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Archetype A. Linear system of three equations, three unknowns. Singular coefficient matrix with dimension 1 null space. Integer eigenvalues and a degenerate eigenspace for coefficient matrix.
Archetype B. System with three equations, three unknowns. Nonsingular coefficient matrix. Distinct integer eigenvalues for coefficient matrix.
Archetype C. System with three equations, four variables. Consistent. Null space of coefficient matrix has dimension 1.
Archetype D. System with three equations, four variables. Consistent. Null space of coefficient matrix has dimension 2. Coefficient matrix identical to that of Archetype E, vector of constants is different.
Archetype E. System with three equations, four variables. Inconsistent. Null space of coefficient matrix has dimension 2. Coefficient matrix identical to that of Archetype D, constant vector is different.
Archetype F. System with four equations, four variables. Nonsingular coefficient matrix. Integer eigenvalues, one has "high" multiplicity.
Archetype G. System with five equations, two variables. Consistent. Null space of coefficient matrix has dimension 0. Coefficient matrix identical to that of Archetype H, constant vector is different.
Archetype H. System with five equations, two variables. Inconsistent, overdetermined. Null space of coefficient matrix has dimension 0. Coefficient matrix identical to that of Archetype G, constant vector is different.
Archetype I. System with four equations, seven variables. Consistent. Null space of coefficient matrix has dimension 4.
Archetype J. System with six equations, nine variables. Consistent. Null space of coefficient matrix has dimension 5.
Archetype K. Square matrix of size 5. Nonsingular. 3 distinct eigenvalues, 2 of multiplicity 2.
Archetype L. Square matrix of size 5. Singular, nullity 2. 2 distinct eigenvalues, each of "high" multiplicity.
Archetype M. Linear transformation with bigger domain than codomain, so it is guaranteed to not be injective. Happens to not be surjective.
Archetype N. Linear transformation with domain larger than its codomain, so it is guaranteed to not be injective. Happens to be onto.
Archetype O. Linear transformation with a domain smaller than the codomain, so it is guaranteed to not be onto. Happens to not be one-to-one.
Archetype P. Linear transformation with a domain smaller that its codomain, so it is guaranteed to not be surjective. Happens to be injective.
Archetype Q. Linear transformation with equal-sized domain and codomain, so it has the potential to be invertible, but in this case is not. Neither injective nor surjective. Diagonalizable, though.
Archetype R. Linear transformation with equal-sized domain and codomain. Injective, surjective, invertible, diagonalizable, the works.
Archetype S. Domain is column vectors, codomain is matrices. Domain is dimension 3 and codomain is dimension 4. Not injective, not surjective.
Archetype T. Domain and codomain are polynomials. Domain has dimension 5, while codomain has dimension 6. Is injective, can't be surjective.
Archetype U. Domain is matrices, codomain is column vectors. Domain has dimension 6, while codomain has dimension 4. Can't be injective, is surjective.
Archetype V. Domain is polynomials, codomain is matrices. Domain and codomain both have dimension 4. Injective, surjective, invertible. Square matrix representation, but domain and codomain are unequal, so no eigenvalue information.
Archetype W. Domain is polynomials, codomain is polynomials. Domain and codomain both have dimension 3. Injective, surjective, invertible, 3 distinct eigenvalues, diagonalizable.
Archetype X. Domain and codomain are square matrices. Domain and codomain both have dimension 4. Not injective, not surjective, not invertible, 3 distinct eigenvalues, diagonalizable.