College Algebra Exam Review 358

College Algebra Exam Review 358 - 368 8. MODULES for a...

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Unformatted text preview: 368 8. MODULES for a variable matrix over ZŒfxi;j g. This in turn implied the identity for arbitrary matrices over an arbitrary commutative ring with identity. Exercises 8.3 8.3.1. Show that the set of multilinear maps is an abelian group under addition. 8.3.2. Show that if ' W M n ! N is multilinear, and 2 Sn , then ' is also multilinear. Show that each of the following sets is invariant under the action of Sn : the symmetric multilinear functions, the skew–symmetric multilinear functions, and the alternating multilinear functions. 8.3.3. (a) (b) (c) Show that .Rn /k has no nonzero alternating multilinear functions with values in R, if k > n. Show that .Rn /k has nonzero alternating multilinear functions with values in R, if k Ä n. Conclude that Rn is not isomorphic to Rm as R–modules, if m ¤ n. 8.3.4. Compute the following determinant by row reduction. Observe that the result is an integer, even though the computations involve rational numbers. 2 3 235 det 44 3 15 3 26 8.3.5. Prove the cofactor expansion identity det.A/ D n X . 1/i Cj ai;j det.Ai;j /: j D1 by showing that the right hand side defines an alternating multilinear function of the columns of the matrix A whose value at the identity matrix is 1. It follows from Corollary 8.3.8 that the right hand is equal to the determinant of A 8.3.6. Prove a cofactor expansion by columns: For fixed j , det.A/ D n X . 1/i Cj ai;j det.Ai;j /: i D1 8.3.7. Prove Cramer’s rule: If A is an invertible n–by–n matrix over R, and b 2 Rn , then the unique solution to the matrix equation Ax D b is given by Q xj D det.A/ 1 det.Aj /; ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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