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adoni [48]
3 years ago
6

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)=3

v−5u, and T2(u)=7v−2u.Find the images of v and u under the composite of T1 and T2.
Mathematics
1 answer:
Doss [256]3 years ago
6 0

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

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