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Exercise:
Let V be a vector space over K with textdimVnin mathbbZ_geq . Then Vcong K^n.

Solution:
Proof. Choose a basis mathcalB for V. Then Phi_mathcalB:Vlongrightarrow K^n is an isomorphism. Does not hold for dimVinfty We've seen that Vcong K^n ntextdimV infty but in general nexists canonical isomorphism V longrightarrow^cong K^n. For example Phi_mathcalB deps on a choice of a basis mathcalB.
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Exercise:
Let V be a vector space over K with textdimVnin mathbbZ_geq . Then Vcong K^n.

Solution:
Proof. Choose a basis mathcalB for V. Then Phi_mathcalB:Vlongrightarrow K^n is an isomorphism. Does not hold for dimVinfty We've seen that Vcong K^n ntextdimV infty but in general nexists canonical isomorphism V longrightarrow^cong K^n. For example Phi_mathcalB deps on a choice of a basis mathcalB.
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dimension, eth, hs22, lineare algebra, proof, vector space
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ENG (English)
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Proof
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