- #1
sanctifier
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Notations:
V denotes a vector space
A, B, C, D denote subspaces of V respectively
≈ denotes the isomorphic relationship of the left and right operand
dim(?) denotes the dimension of "?"
Question:
Find a vector space V and decompositions:
V = A ⊕ B = C ⊕ D
with A≈C but B and D are not isomorphic.
My opinion:
dim(V)=dim(A)+dim(B)=dim(C)+dim(D) and dim(A)=dim(C), but dim(B)≠dim(D) since V may not be finite-dimensional. It's an idea not an example, would you make a concrete example of V?
Thanks for any help!
V denotes a vector space
A, B, C, D denote subspaces of V respectively
≈ denotes the isomorphic relationship of the left and right operand
dim(?) denotes the dimension of "?"
Question:
Find a vector space V and decompositions:
V = A ⊕ B = C ⊕ D
with A≈C but B and D are not isomorphic.
My opinion:
dim(V)=dim(A)+dim(B)=dim(C)+dim(D) and dim(A)=dim(C), but dim(B)≠dim(D) since V may not be finite-dimensional. It's an idea not an example, would you make a concrete example of V?
Thanks for any help!