Dual map
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Exercise:
Let VW be vector spaces over K. Let T:Vlongrightarrow W be a linear map. Deifne a new map T^*:W^*longrightarrow V^* T^*l:lcirc T forall lin W^* i.e. l:Wlongrightarrow K. Then T^* is a linear map. It is called the dual map to T.
Solution:
Proof. Let lin W^* i.e. l:Wlongrightarrow K linear. Clearly lcirc T:V longrightarrow K is a linear map because both T l are so Tl: lcirc Tin V^*. This shows T^*:W^*longrightarrow V^* has its target indeed V^* as claimed. We'll show now that T^* is a linear map. Let lin W^* alpha in K. We have that T^*alpha lin V^* Longrightarrow T^*alpha lvalpha lcirc Tv alpha lTv alpha lTv alpha lcirc Tv alpha T^*lv &Longrightarrow T^*alpha l alpha T^*l Similarly one shows T^*l_+l_T^*l_+T^*l_.
Let VW be vector spaces over K. Let T:Vlongrightarrow W be a linear map. Deifne a new map T^*:W^*longrightarrow V^* T^*l:lcirc T forall lin W^* i.e. l:Wlongrightarrow K. Then T^* is a linear map. It is called the dual map to T.
Solution:
Proof. Let lin W^* i.e. l:Wlongrightarrow K linear. Clearly lcirc T:V longrightarrow K is a linear map because both T l are so Tl: lcirc Tin V^*. This shows T^*:W^*longrightarrow V^* has its target indeed V^* as claimed. We'll show now that T^* is a linear map. Let lin W^* alpha in K. We have that T^*alpha lin V^* Longrightarrow T^*alpha lvalpha lcirc Tv alpha lTv alpha lTv alpha lcirc Tv alpha T^*lv &Longrightarrow T^*alpha l alpha T^*l Similarly one shows T^*l_+l_T^*l_+T^*l_.