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https://texercises.com/exercise/dimension-injectivity-surjectivity/
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Exercise:
Let T:Vlongrightarrow W be a linear map where VW are finite dimensional vector spaces. The following holds: abcliste abc If textdimW textdimV then T is NOT injective. abc If textdimW textdimV then T is NOT surjective. abc If If textdimW textdimV then T is bijective iff T is injective iff T is surjective. abcliste

Solution:
Proof. abcliste abc Since textImT is a subspace of W we have textdim ImT leq textdimW textdimV By the previous theorem rank theorem we have textdim KerTtextdimV-textdim ImT &Longrightarrow textKerTneq Longrightarrow by a previous Proposition T is not injective. abc textdim ImTtextdimV-textdim KerTleq textdimV textdim W &Longrightarrow textImTsubsetneq W By a previous proposition T is not surjective. abc Tquad textis injective &iff textKerT &iff textdim KerT &iff textdimVtextdim ImT &iff textdimWtextdim ImT &iff textImTW &iff Tquad textis surjective. abcliste
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Exercise:
Let T:Vlongrightarrow W be a linear map where VW are finite dimensional vector spaces. The following holds: abcliste abc If textdimW textdimV then T is NOT injective. abc If textdimW textdimV then T is NOT surjective. abc If If textdimW textdimV then T is bijective iff T is injective iff T is surjective. abcliste

Solution:
Proof. abcliste abc Since textImT is a subspace of W we have textdim ImT leq textdimW textdimV By the previous theorem rank theorem we have textdim KerTtextdimV-textdim ImT &Longrightarrow textKerTneq Longrightarrow by a previous Proposition T is not injective. abc textdim ImTtextdimV-textdim KerTleq textdimV textdim W &Longrightarrow textImTsubsetneq W By a previous proposition T is not surjective. abc Tquad textis injective &iff textKerT &iff textdim KerT &iff textdimVtextdim ImT &iff textdimWtextdim ImT &iff textImTW &iff Tquad textis surjective. abcliste
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dimension, eth, hs22, injective, lineare algebra, proof, vector space
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ENG (English)
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