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Constructing new vector spaces out of old ones

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Exercise

https://texercises.com/exercise/constructing-new-vector-spaces-out-of-old-ones/

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Exercise:
Let VW be two vector spaces over K. Then textHomVW has the structure of a vector space over K if we ow it with the following operations: forall T_ T_in textHomVW T_+T_v&:T_v+T_v forall vin V forall Tin textHomVW alpha in K alpha Tv&:alpha Tv forall vin V

Solution:
Proof. We claim that T_+T_in textHomVW i.e. T_+T_ is a linear map. Indeed T_+T_av T_av+T_avquad textlin. aT_v+aT_v aT_v+T_c aT_+T_vquad textdef. T_+T_v_+v_quad textdef. T_v_+v_+T_v_+v_quad textlin. T_v_+T_v_+T_v_+T_v_quad textdef. T_+T_v_+T_+T_v_ This shows T_+T_ is a linear map. The proof that alpha T is linear is similar. So far we have defined +: textHomVWtimes textHomVW longrightarrow textHomVW : Ktimes textHomVW longrightarrow textHomVW. bf Neutral element. The map :Vlongrightarrow W vmapsto forall vin V is linear. We claim that +TT. Indeed +Tvv+Tv+TvTv forall vin V. Similarly one shows that T+T. Now it remains to be proven that the eight axioms that define a vector space hold. Example distributivity: forall alpha in K T_ T_ in textHomVW: alpha T_+T_alpha T_+alpha T_ Indeed forall vin V: alphaT_+T_valpha T_+T_v alphaT_v+T_v alpha T_v+alpha T_v alpha T_v+alpha T_v alpha T_+alpha T_v &Longrightarrow alpha T_+T_alpha T_+alpha T_

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Tags	eth, hs22, lineare algebra, proof, vector space
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Difficulty	(3, default)
Points	0 (default)
Language	(English)
Type	Proof
Creator	rk
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