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Isomorphic linear space

by bernoli123
Tags: isomorphic, linear, space
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bernoli123
#1
Apr30-11, 03:45 AM
P: 11
two linear spaces S and S1 over F are isomorphic if and only if there is a one-to-one
correspondence x↔ x1 between the elements x [tex]\in[/tex] S and x1 [tex]\in[/tex] S1
such that if x ↔ x1 and y ↔ y1 then x+y ↔ x1+y1 and ax ↔ ax1
(y [tex]\in[/tex] S , y1 [tex]\in[/tex] S1, a [tex]\in[/tex] F).
prove that two finite -dimensional spaces are isomorphic if and only if they are of the same dimension.
(The correspondence or mapping defining isomorphic linear spaces is called an
isomorphism).
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Fredrik
#2
Apr30-11, 05:41 AM
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Another way to say that, is that a vector space isomorphism is a linear bijection. So if U and V are vector spaces, and dim U=dim V=n (where n is some positive integer), you need to find a linear bijection T:U→V. I suggest that you use a basis for U and a basis for V to define a function T:U→V, and then show that T is injective, surjective, and linear.

If you want more help, you need to show us your attempt.


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