Math 260, Spring 2017, Assignment 12

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

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

1. Transpose (of a matrix).
2. Orthogonal matrix.
3. $n$-pattern.
4. Inversion (in an $n$-pattern).
5. Sign (of an $n$-pattern).
6. Determinant (of an $n\times n$ matrix).

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

1. Theorem concerning $QR$ decomposition.
2. Criterion for orthonormal columns, in terms of the transpose.
3. Formula for $2\times2$ determinants.
4. Sarrus' rule (for $3\times3$ determinants).
5. Theorem concerning the total number of $n$-patterns.
6. Lemma relating the signs of $\pi$ and $\sigma$ when $\sigma$ is obtained from $\pi$ by a single row swap.

Solve the following problems:

1. Section 5.2, problems 5 and 13 (you have already done the Gram-Schmidt process in these problems last week; now find and verify the $QR$ decompositions).
2. Section 5.3, problems 1, 3, and 5.
3. Section 6.1, problems 1, 3, 5, 7, 11, 13, 15, 43, 44, and 45.

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