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, $v,u\in V$

$T_1:V\rightarrow V$

$T_2:V\rightarrow V$

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 $T_2$

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

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

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

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

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

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

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