Homework7

# Homework7 - ﬁeld Z 7(a Factor the polynomials x 2 x 1 and...

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Homework 7 Math 332, Spring 2010 These problems must be written up in L A T E X, and are due this Thursday, May 6. 1. (a) Let Z ± 2 ² = { a + b 2 : a,b Z } . Prove that Z ± 2 ² is a subring of the real numbers. (b) Let M 2 ( R ) be the ring of 2 × 2 matrices with real entries, and let S = ³´ a 2 b b a µ : a,b Z . Prove that S is a subring of M 2 ( R ). (c) Prove that S is isomorphic to Z ± 2 ² . 2. Much of the algebra and linear algebra you know works perfectly well over any ﬁeld. The following exercises ask you to perform certain familiar computations over the ﬁnite
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Unformatted text preview: ﬁeld Z 7 . (a) Factor the polynomials x 2 + x + 1 and x 3 + 2 x + 5, where the coeﬃcients are elements of Z 7 . (b) Solve the following system of linear equations, where x and y are elements of Z 7 : 5 x + 2 y = 1, 3 x + 5 y = 5 . (c) Find the inverse of the matrix ´ 3 2 4 1 µ over Z 7 . (d) Find the eigenvalues of the matrix ´ 2 4 2 2 µ over Z 7 . For each eigenvalue, ﬁnd a corresponding eigenvector....
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