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Prove. Let U, V and W be vector spaces over the same field F. Let P: U V and Q: V W be linear mappings from U into V and V into W respectively. Then if functions P and Q are linear, the product QP is also linear.

Proof. For any vectors v, w in V and scalars a, b in F,

(Q P)(av + bw) = Q(P(av + bw)) = Q(aPv + bPw) = aQ(Pv) + bQ(Pw) = a (Q P)v + b (Q P)w

Thus Q P is linear.

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