I was studing linear algebra, being more specific linear transformation, I think that a composite function is a linear transformation, but how to prove it?
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$\begingroup$ Please specify what you mean by a "composite function". $\endgroup$Eli Rose– Eli Rose2015-06-06 04:40:56 +00:00Commented Jun 6, 2015 at 4:40
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$\begingroup$ Presumably the composition of two linear transformations. What is your definition of a linear transformation? What happens when you plug the results of one transformation, into another? $\endgroup$pjs36– pjs362015-06-06 04:43:28 +00:00Commented Jun 6, 2015 at 4:43
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$\begingroup$ Let $f:X\to Y$ and $g:Y\to Z$ be linear maps/transformations, then $f\circ g:X\to Z$ is again a linear map. By definition of a linear map you need to proof that: $f(g(x+y))=f(g(x))+f(g(y))$ and $f(g(ax))=af(g(x))$ for any scalar a. $\endgroup$John– John2015-06-06 04:50:48 +00:00Commented Jun 6, 2015 at 4:50
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1 Answer
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If $f: A \to B, g: B \to C$ are $2$ linear transformations we can show that $g\circ f: A \to C$ is also a linear transformation. To this end, let $x,y \in A$ and note that $$(g\circ f)(x+y) = g(f(x+y)) = g(f(x)+f(y)) = g(f(x))+g(f(y)=(g\circ f)(x)+(g\circ f)(y)$$and $$(g\circ f)(rx)=g(f(rx))=g(rf(x))=rg(f(x))=r(g\circ f)(x).$$ These identities show the composition is linear.
