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I want an "algebraic" (something that does not use matrices directly ) proof of the epic monic factorisation property of linear maps . It would be nice if I get to see a proof (with some motivation) of the following problem :

Let T : V1 → V2 be a linear transformation. Show that there is a vector space V and linear maps R : V1 → V, S : V → V2 such that R is one to one, S is onto and T = S ◦ R.

John
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1 Answers1

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Let $V = V_1 \oplus V_2$, and let $R(v) = (v, T(v))$, and let $S(v_1, v_2) = v_2$. Then $R$ is one-to-one, $S$ is onto, and $S(R(v)) = S(v, T(v)) = T(v)$.

Christopher
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  • Here , is the sum V1+V2 direct ? If so , please explain a bit more . – John Oct 18 '18 at 14:47
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    @John yes, $\oplus$ means direct sum, i.e. the set of pairs $(v_1, v_2)$ where $v_1 \in V_1$, $v_2 \in V_2$; the direct sum of two vector spaces is also a vector space. – Christopher Oct 18 '18 at 14:51
  • Is it true that any vector space V can be written as a direct sum of two vector spaces ? If yes, then can you please refer me to some reading on that part ! Thanks a lot for your answer ! – John Oct 18 '18 at 14:53
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    @John why do you ask, that doesn't seem related at all? – Christopher Oct 18 '18 at 14:55
  • I thought that the fact that we can write V as direct sum of two vector spaces V1 and V2 needed some justification . So , I asked the above question.Edit (below) - Oops ! thank you for explaining everything with such patience :P – John Oct 18 '18 at 14:56
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    @John we are defining $V$ to be the direct sum $V_1 \oplus V_2$. Remember the question asks to show there exists a $V$ with the required properties, we weren't given a $V$ to start with. – Christopher Oct 18 '18 at 14:58
  • Christopher ,V1 and V2 should have singleton intersection for their sum to be direct ? Please correct me . – John Oct 24 '18 at 16:38
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    @John there are two similar but not-quite-the-same notions of direct sum, internal direct sum and external direct sum. See this answer for an explanation of the difference https://math.stackexchange.com/questions/1507156/for-direct-sum-of-modules-can-the-two-modules-be-the-same/1507179#1507179 In the answer I gave, we use the external direct sum. – Christopher Oct 25 '18 at 08:57