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https://texercises.com/exercise/dimension-of-quotient-space/
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Exercise:
Suppose that V is finite dimension. Then V/U is finite dimensional too and textdimV/UtextdimV-textdimU.

Solution:
Proof. We have seen that pi:Vlongrightarrow V/U is linear and surjective Longrightarrow V/U is finite dimensional always if T:W'longrightarrow W'' is surjective linear map and W' is finite dimensional then W'' is also finite dimension because if w_'...w_m' is a basis for W' then Tw_'...Tw_m' span textImTW''. We now apply the rank theorem for pi: textdimVtextdim Kerpi+textdim Impi. But textKerpiU textImV/U &Longrightarrow textdimV/U textdimV-textdimU
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Exercise:
Suppose that V is finite dimension. Then V/U is finite dimensional too and textdimV/UtextdimV-textdimU.

Solution:
Proof. We have seen that pi:Vlongrightarrow V/U is linear and surjective Longrightarrow V/U is finite dimensional always if T:W'longrightarrow W'' is surjective linear map and W' is finite dimensional then W'' is also finite dimension because if w_'...w_m' is a basis for W' then Tw_'...Tw_m' span textImTW''. We now apply the rank theorem for pi: textdimVtextdim Kerpi+textdim Impi. But textKerpiU textImV/U &Longrightarrow textdimV/U textdimV-textdimU
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dimension, eth, hs22, lineare algebra, proof, quotient space
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(3, default)
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ENG (English)
Type
Proof
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