Unformatted text preview: trix is nonzero, and that the determinant satisﬁes the identity det .AB/ D det .A/ det .B/ . Therefore, det W GL .n; R / ! R ± is a homomorphism from the group of invertible matrices to the group of nonzero real numbers under multiplication. Example 2.4.6. Recall that a linear transformation T W R n ! R n has the property that T. a C b / D T. a / C T. b / . Thus T is a group homomorphism from the additive group R n to itself. More concretely, for any n-by-n matrix M , we have M. a C b / D M a C M b . Thus multiplication by M is a group homomorphism from the additive group R n to itself. Example 2.4.7. Let G be any group and a and element of G . The the map from Z to G given by k 7! a k is a group homomorphism. This is equivalent to the statement that a k C ` D a k a ` for all integers k and ` . The image of this homomorphism is the cyclic subgroup of G generated by g ....
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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.
- Fall '08