Let V and W be two vector spaces an d v epsilon V a vector. Define a map ev: (V1

Question

Let V and W be two vector spaces an d v epsilon V a vector. Define a map ev: (V1 W) rightarrow W by ev(T) = T(v). Prove that ev is a linear map.

Explanation / Answer

Proof: Step1: Let v in V and V and W are two vector spaces over field F. Define ev: L(V,W)----> W by ev(T)=T(v). Here L(V,W) is the space of all linear transformations form V in to W. So, given v in V and T in L(V,W) then T(v) in W. Hence ev map is well defined. Step2: Claim: ev is a linear map. Let T, S in L(V,W) and a in F. Then ev(aT+S)=(aT+S)(v), by definition = (aT)(v)+S(v) =a T(v) +S(v) =a ev(T)+S(v). So, we have ev(aT+S)=a ev(T)+ev(S) for any T, S in L(V,W) and a in F. Hence ev is a linear map. Hence the proof.

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