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Consider two linear transformations y = T(x) and z = L(y), where T goes from R^m to R^p and L goes from R^p to R^n. Is the transformationz = L(T(x)) linear as well ? [The transformation z = L(T(x)) is called the composite of T and L.]



Answer :

Yes.

Proof: Consider x, y in R^m. Then since T is linear, we have:

T(a*x + b*y) = a*T(x) + b*T(y)

But since L is linear, we have:

L(a*T(x) + b*T(y)) = a*L(T(x)) + b*L(T(y))

So:

L(T(a*x + b*y)) = a*L(T(x)) + b*L(T(y))

and the composition is linear.