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Isomorphic linear space 
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#1
Apr3011, 03:45 AM

P: 11

two linear spaces S and S1 over F are isomorphic if and only if there is a onetoone
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). 


#2
Apr3011, 05:41 AM

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PF Gold
P: 9,421

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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