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Question: 25 please...

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v, v2, b. 24. Prove that if T:R-Rm is a one-to-one linear transto e linear transformation and S of linearly independent vectors from R Hint: Begin the proof by considering the dependence test equa e lefit side using the lincarity properties of T and use the Kernel Test for Injectivity 31. S ,v, 25. Use the previous Exercise to prove that if T : R-Ris one to one and S = { , , is any basis for R, then T , ), TTE), . . . , TT,) } is a basis for range(7). a. What is the ambient space of range(Ti)? b. What is the ambient space of ker(T2)? Prove that T2。7,-Znm if and only if range(T 1) g: ker(T2). Recall that Zn m : R → Rm is the zero transformation, where Zn,m(V) = 0 m for all R Notc b G and (b), bolh sabspaces range() and ker() are in R, so requi one to be a subset of the other is a possibility. Suppose that m 32. d. n. State and prove an analogouo 25 please
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