- #1
math_grl
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So this is supposed be an introductory problem for tensor products that I was trying to do to verify I am understanding tensor products...turns out I'm not so much
Show that [tex]M_n(K)[/tex] is isomorphic as an [tex]F[/tex]-algebra to [tex]K \otimes_F M_n(F)[/tex] where [tex]F[/tex] is a field and [tex]K[/tex] is an extension field of [tex]F[/tex] and [tex]M_n(K)[/tex] means all the nxn matrices that have entries in K.
So I figure as F-algebras we need to show that we have a ring homomorphism that is linear (preserving the scalar multiplication) or showing they are isomorphic as F-modules (vec. sp's) then showing preservation of the multipication. Either way, my attempts are fruitless.
Show that [tex]M_n(K)[/tex] is isomorphic as an [tex]F[/tex]-algebra to [tex]K \otimes_F M_n(F)[/tex] where [tex]F[/tex] is a field and [tex]K[/tex] is an extension field of [tex]F[/tex] and [tex]M_n(K)[/tex] means all the nxn matrices that have entries in K.
So I figure as F-algebras we need to show that we have a ring homomorphism that is linear (preserving the scalar multiplication) or showing they are isomorphic as F-modules (vec. sp's) then showing preservation of the multipication. Either way, my attempts are fruitless.