By Steven Roman
ISBN-10: 0387728287
ISBN-13: 9780387728285
This graduate point textbook covers a particularly wide diversity of subject matters. The booklet first deals a cautious dialogue of the fundamentals of linear algebra. It then proceeds to a dialogue of modules, emphasizing a comparability with vector areas, and offers a radical dialogue of internal product areas, eigenvalues, eigenvectors, and finite dimensional spectral concept, culminating within the finite dimensional spectral theorem for regular operators. the hot version has been revised and encompasses a bankruptcy at the QR decomposition, singular values and pseudoinverses, and a bankruptcy on convexity, separation and optimistic recommendations to linear structures.
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Sample text
Fortunately, principal ideal domains have this property. 30 Every principal ideal domain 9 is a unique factorization domain. Proof. Let 9 be a nonzero nonunit. If is irreducible, then we are done. If not, then ~ , where neither factor is a unit. If and are irreducible, we are done. If not, suppose that is not irreducible. Then ~ , where neither nor is a unit. Continuing in this way, we obtain a factorization of the form (after renumbering if necessary) ~ ~ ² ³ ~ ² ³² ³ ~ ² ³² ³ ~ Ä Each step is a factorization of into a product of nonunits.
Proof. Let 9 be a nonzero nonunit. If is irreducible, then we are done. If not, then ~ , where neither factor is a unit. If and are irreducible, we are done. If not, suppose that is not irreducible. Then ~ , where neither nor is a unit. Continuing in this way, we obtain a factorization of the form (after renumbering if necessary) ~ ~ ² ³ ~ ² ³² ³ ~ ² ³² ³ ~ Ä Each step is a factorization of into a product of nonunits. However, this process must stop after a finite number of steps, for otherwise it will produce an infinite sequence Á Á Ã of nonunits of 9 for which b properly divides .
Now we repeat the process, moving # from the second list to the first list # Á # Á Á Ã Á Â # Á Ã Á # As before, the vectors in the first list are linearly dependent, since they spanned = before the inclusion of # . However, since the # 's are linearly independent, any nontrivial linear combination of the vectors in the first list that equals must involve at least one of the 's. Hence, we may remove that vector, which again by reindexing if necessary may be taken to be and still have a spanning set # Á # Á Á Ã Á Â # Á Ã Á # Once again, the first set of vectors spans = and the second set is still linearly independent.
Advanced Linear Algebra (Graduate Texts in Mathematics, Volume 135) by Steven Roman
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