Math 260, Spring 2017, Assignment 12

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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:[edit]

  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×n matrix).

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

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

Solve the following problems:[edit]

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

Questions:[edit]

Solutions:[edit]