problemset07

problemset07 - (4) Show that t R n ⊗ M n ( R ) R n ∼ =...

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June 4, 2008 Math 220C Problem Set #7 Due: Friday, June 6 (but extensions will be available.) Let R be a ring with 1, and let M n ( R ) be the ring of n × n matrices with entries in R . The set R n of column vectors of length n is then an ( M n ( R ) ,R )-bimodule, and its transpose t R n (the set of row vectors of length n ) is an ( R,M n ( R ))-bimodule. (1) Show that any n × n matrix with entries in R can be written as a sum of rank 1 matrices. (2) Given M M n ( R ), show that there exist v 1 ,...,v k R n and t w 1 ,..., t w k t R n such that M = v 1 t w 1 + ··· + v k t w k . (3) Conclude that R n R t R n = M n ( R ), an isomorphism of ( M n ( R ) ,M n ( R ))- bimodules.
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Unformatted text preview: (4) Show that t R n ⊗ M n ( R ) R n ∼ = R , an isomorphism of ( R,R )-bimodules. (5) Show the following isomorphisms of R-algebras: (a) C ⊗ R C ∼ = C ⊕ C (b) H ⊗ R C ∼ = M 2 ( C ) (c) H ⊗ R H ∼ = M 4 ( R ) where R , C , and H denote the real numbers, the complex numbers, and the quaternions, respectively. (6) (The exercise on the “Clifford Algebras” handout). Show that Cliff (2 , 0) ∼ = M 2 ( R ) and that Cliff (0 , 2) ∼ = H ....
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This note was uploaded on 08/06/2008 for the course MATH 220 taught by Professor Morrison during the Spring '08 term at UCSB.

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