Question
An n-dimensional linear operator over an algebraically closed field must have n of these values by the fundamental theorem of algebra. For 10 points each:
[10m] Name these values that are not generally diagonal entries but sum up to the trace of an operator. For an operator M, these values appear on the main diagonal in the diagonalization of M.
ANSWER: eigenvalues [reject “eigenvectors”]
[10h] For two linear operators with eigenvalues a and b, a times b is an eigenvalue of this operation applied to the operators. Applying this operation to vector spaces of dimensions m and n generates one with dimension m times n.
ANSWER: tensor product [accept Kronecker product; prompt on outer product; reject “Cartesian product” or “direct product”]
[10e] Tensor products are used to prove that the set of numbers arising as roots to polynomials over this set is itself a ring. This set is denoted with a blackboard capital Z and, unlike the natural numbers, is closed under subtraction.
ANSWER: integers [accept algebraic integers]
<Other Science>
Summary
| 2023 ACF Winter @ Columbia | 11/11/2023 | Y | 4 | 15.00 | 75% | 75% | 0% |
Data
| Rutgers A | Columbia A | 10 | 0 | 10 | 20 |
| Columbia B | Cornell C | 10 | 0 | 10 | 20 |
| Haverford | Rowan A | 0 | 0 | 0 | 0 |
| NYU A | Yale A | 10 | 0 | 10 | 20 |