Math 260, Spring 2017, 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. Eigenvalue.
2. Eigenvector.
3. $\lambda$-eigenspace.
4. Characteristic polynomial (of a square matrix $A$).

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

1. Laplace expansion (of a determinant, in terms of smaller determinants).

Solve the following problems:

1. Section 6.2, problem 8 (please use Laplace expansion in order to gain familiarity with that method).
2. Section 7.2, problems 1, 3, 5, 7, 9, 11, and 13.
3. Section 7.3, problems 1, 3, 5, 7, 9, and 11 (here "diagonalize" means "find the matrix of the transformation in the coordinate system defined by the basis you have computed").

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