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Let V be a vector space, v,u∈V, and let T1:V→V and T2:V→V be linear transformations such that T1(v)=3v+2u, T1(u)=−7v+6u, T2(v)=3v−5u, and T2(u)=7v−2u.Find the images of v and u under the composite of T1 and T2.

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Answer with Step-by-step explanation:

We are given that V be a vector space, [tex]v,u\in V[/tex]

[tex]T_1:V\rightarrow V[/tex]

[tex]T_2:V\rightarrow V[/tex]

be linear transformation such that

[tex]T_1(v)=3v+2u[/tex]

[tex]T_1(u)=-7v+6u[/tex]

[tex]T_2(v)=3v-5u[/tex]

[tex]T_2(u)=7v-2u[/tex]

We have to find the images of v and u under the composite of [tex]T_1[/tex] and [tex]T_2[/tex]

[tex]T_2T_1(u)=T_2(T_1(u)=T_2(-7v+6u)=-7T_2(v)+6T_2(u)[/tex]

[tex]T_2T_1(u)=-7(3v-5u)+6(7v-2u)=-21v+35u+42v-12u=21v+23u[/tex]

[tex]T_2T_1(u)=21v+23u[/tex]

[tex]T_2T_1(v)=T_2(T_1(v))=T_2(3v+2u)=3T_2(v)+2T_2(u)[/tex]

[tex]T_2T_1(v)=3(3v-5u)+2(7v-2u)[/tex]

[tex]T_2T_1(v)=9v-15u+14v-4u[/tex]

[tex]T_2T_1(v)=23v-19u[/tex]

Answer with Step-by-step explanation:

We are given that V be a vector space,

be linear transformation such that

We have to find the images of v and u under the composite of  and

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