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created: 2021-11-14 12:30:42
modified: 2022-01-10 04:13:04
Statement: Let and be finite-dimensional vector spaces of the same dimension.
Then . (they are isomorphic)
This implies that any -dimensional vector space over the field is essentially since we get an isomorphism once we choose a basis for .
Proof
Let be a basis for and let be a basis for . There is a linear map such that for . Note that is surjective since spans .
Now the rank theorem implies that , so is also Injective.
Hence is an isomorphism.