Math 260, Fall 2015, Assignment 14

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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

1. Eigenvector (of a linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$).
2. Eigenvalue.
3. Eigenspace (associated with a particular eigenvalue).
4. Eigenbasis.
5. Characteristic polynomial.
6. Characteristic equation.
7. Algebraic multiplicity (of an eigenvalue; this is defined carefully on page 310 of the text).
8. Geometric multiplicity.
9. Diagonal matrix.
10. Diagonalization (of a transformation $T$).

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

1. Theorem relating algebraic multiplicity to geometric multiplicity (this is Fact 7.3.7 on page 324).
2. Theorem concerning the form of the matrix $[T]_{\mathcal{B}}$ when $\mathcal{B}$ is an eigenbasis for $T$.

Solve the following problems:

1. Section 7.2, problems 1, 3, 5, 13, and 15.
2. Section 7.3, problems 1, 3, 5, 13, and 19.
3. Section 7.4, problems 1, 3, and 5.

--------------------End of assignment--------------------