To find: The linear transformation (T_2T_1)(v) for an arbitrary vector v in V. The vectors {v_1,v_2} is vasis for the vector space V. Given: The linea

Emily-Jane Bray 2021-02-09 Answered
To find:
The linear transformation (T2T1)(v) for an arbitrary vector v in V.
The vectors {v1,v2} is vasis for the vector space V.
Given:
The linear transformation with satisfying equations T1(v1)=3v1+v2,
T1(v1)=3v1+v2, T2(v1)=5v2, and T2(v2)=v1+6v2 are given as
T1 : V V and T2 : VV.
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Expert Answer

Laaibah Pitt
Answered 2021-02-10 Author has 98 answers
Calculation:
To solve for T2(T1(v)) find T1(v) use properties of linear transformation.
T1(v)=T1(av1+bv2)
T1(v)=aT1(v1)+bT1(v2)
Substitute 3v1+v2 for T1(v1) and 0 for T1(v2) in the above expression.
T1(v)=a(3v1+v2)+b(0)
=a(3v1+v2)
Substitute a(3v1+v2) for T1(v) to find T2(T1(v)) and it is written as,
T2(T1(v))=T2(a(3v1+v2))
T2(T1(v))=3aT2(v1)+aT2(v2)
Substitute 5v2 for T2(v1) and v1+6v2 for T2(v2) in the above expression.
T2(T1(v))=3a(5v2)+a(v1+6v2)
15av2av1+6av2
=av1  9av2
Therefore, the linear transformation (T2T1)(v) is
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