Math 260, Fall 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

Read:

1. Section 7.1.
2. Section 7.2.
3. Section 7.3.

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

1. $\mathrm{adj}(A)$ (i.e. the classical adjoint of $A$).
2. Eigenvector (of a square matrix $A$).
3. Eigenvalue (of a square matrix $A$).
4. Eigenbasis (of $\mathbb{R}^n$, with respect to an $n\times n$ matrix $A$).
5. $E_{\lambda}$ (i.e. the $\lambda$-eigenspace of $A$).
6. Characteristic equation (for the eigenvalues of $A$).

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

1. Cramer's Rule.
2. Formula for the inverse of a square matrix $A$ (in terms of the determinant and the adjoint of $A$).
3. Theorem relating $E_{\lambda}$ to kernels.

Solve the following problems:

1. Section 6.3, problems 23, 24, 31, 33, 34, and 35.
2. Section 7.1, problems 1, 2, 3, 4, 5, 6, 15, 19, 21, 25, 27, and 29.
3. Section 7.2, problems 1, 3, 5, 7, 9, 11, and 13 (see p. 311 for the definition of "algebraic multiplicity").
4. Section 7.3, problems 1, 3, 5, 7, 9, 11, and 13 (see p. 341 for the definition of "geometric multiplicity").

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