Math 260, Fall 2015, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Carefully define the following terms, then give one example and one non-example of each:

1. Transpose (of an $n\times m$ matrix).
2. Classical adjoint (of a square matrix $A$).
3. State vector (of a linear discrete dynamical system).
4. State space (of a linear discrete dynamical system).
5. Evolution operator (of a linear discrete dynamical system).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning three essential properties of the determinant.
2. Theorem concerning uniqueness of functions with the properties referenced above.
3. Theorem concerning Laplace expansion (of a determinant across an arbitrary row or down an arbitrary column).
4. Theorem relating invertibility of a square matrix to non-vanishing of its determinant.
5. Theorem concerning the determinant of the transpose.
6. Theorem concerning the determinant of the inverse.
7. Theorem concerning determinants of similar matrices.
8. Cramer's Rule.
9. Formula for the inverse of an $(n\times n)$ invertible matrix.

Solve the following problems:

1. Section 6.2, problems 1, 3, 7, 11, 13, and 15.
2. Section 6.3, problems 23, 25, and 26.

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