10. Let U, V and W be vector spaces. Let S U right arrow V and T.pdf

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10. Let U, V and W be vector spaces. Let S: U right arrow V and T : V right arrow W be linear transformations. Define R: U right arrow W by letting R(u) = T(S(u)) for all u in U. (1) Show that R is a linear transformation. (2) Show that if S and T are one-to-one, then R is one-to-one. Solution (1) R(au+v) = T(S(au+v))=T(aS(u)+S(v))=T(aS(u))+T(S(v))=aT(S(u))+T(S(v)) =aR(u)+R(v). (2) Let R(u)=R(v) T(S(u))=T(S(v)) Since T is one-to-one S(u)=S(v) Since S is one-to-one u=v. R is one-to-one.

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