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Exercise:
Two finite dimensional vector spaces V and W are isomorphic iff textdimVtextdimW.

Solution:
Proof of Longrightarrow. Suppose T:Vlongrightarrow W is an isomorphism. By a previous proposition textKerT and textImTW. By the rank theorem we have textdimWtextdim ImTtextdimV-textdim KerT textdimV Proof of Longleftarrow. Ase textdimVtextdimW and denote this dim by n. Let v_...v_n be a basis for V and w_...w_n a basis for W. Define a linear map T:Vlongrightarrow W by Tv_iw_iquad forall leq i leq n. By a previous theorem such a linear map exists. By the rank theorem textdim KerTn-textdim ImTn-n Longrightarrow T is also injective Longrightarrow T is an isomorphism.
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Exercise:
Two finite dimensional vector spaces V and W are isomorphic iff textdimVtextdimW.

Solution:
Proof of Longrightarrow. Suppose T:Vlongrightarrow W is an isomorphism. By a previous proposition textKerT and textImTW. By the rank theorem we have textdimWtextdim ImTtextdimV-textdim KerT textdimV Proof of Longleftarrow. Ase textdimVtextdimW and denote this dim by n. Let v_...v_n be a basis for V and w_...w_n a basis for W. Define a linear map T:Vlongrightarrow W by Tv_iw_iquad forall leq i leq n. By a previous theorem such a linear map exists. By the rank theorem textdim KerTn-textdim ImTn-n Longrightarrow T is also injective Longrightarrow T is an isomorphism.
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dimension, eth, hs22, isomorphism, lineare algebra, proof, vector space
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ENG (English)
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