Question

Let U, V and W are vector spaces over the same field K. Prove that for the linear transformations, F : U → V and G : V → W, composite function G°F is a linear map from U to W.

Answer 1

SOLUTION

TO PROVE

Let U, V and W are vector spaces over the same field K. Prove that for the linear transformations, F : U → V and G : V → W, composite function GoF is a linear map from U to W.

PROOF

Let α , β ∈ U and c ∈ K

Now

GoF(α + β)

= G[ F(α + β) ] (By property of composition)

= G[ F(α) + F(β) ] ( Since F is linear )

= G[ F(α) ] + G [F(β) ] ( Since G is linear )

= GoF(α) + GoF(β)

Again

GoF(cα)

= G[ F(cα) ] (By property of composition)

= G[ cF(α)] ( Since F is linear )

= cG[ F(α) ] ( Since G is linear )

= c GoF(α)

Since α , β is arbitrary

So GoF is a linear map from U to W.

Hence the proof follows

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