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.
10. Let U, V and W be vector spaces. Let S U right arrow V and T.pdf
1. 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
