Opgave 4a. Explain that \(\prod_{i\in I}V_i\) has a unique structure as \(k\)-vector space such tha the projections \(L_j:\bigoplus_{i\in I}V_i \to V_j : (v_i)_{i\in I}\mapsto v_j\) are \(k\)-linear for all \(j\in I\).
Bewijs. \[L_j((v_i)_{i\in I} +(w_i)_{i \in I})=L_j((v_i+w_i)_{i \in I}) = v_j+w_j =L_j((v_i)_{i\in I})+L_j((w_i)_{i\in I} )\]