Question

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.

Answer 1

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

Answer 2

Answer with Step-by-step explanation:

We are given that V be a vector space,
`v,u\in V`

`T_1:V\rightarrow V`

`T_2:V\rightarrow V`

be linear transformation such that

`T_1(v)=3v+2u`

`T_1(u)=-7v+6u`

`T_2(v)=3v-5u`

`T_2(u)=7v-2u`

We have to find the images of v and u under the composite of
`T_1` and
`T_2`

`T_2T_1(u)=T_2(T_1(u)=T_2(-7v+6u)=-7T_2(v)+6T_2(u)`

`T_2T_1(u)=-7(3v-5u)+6(7v-2u)=-21v+35u+42v-12u=21v+23u`

`T_2T_1(u)=21v+23u`

`T_2T_1(v)=T_2(T_1(v))=T_2(3v+2u)=3T_2(v)+2T_2(u)`

`T_2T_1(v)=3(3v-5u)+2(7v-2u)`

`T_2T_1(v)=9v-15u+14v-4u`

`T_2T_1(v)=23v-19u`