Use a counterexample to show that the statement is false. {T}:{R}^{2}to{R}^{2},{T}{left({x}_{{2}},{x}_{{2}}right)}={left({x}_{{1}}+{4},{x}_{{2}}right)} is a linear transformation?

Anish Buchanan 2021-01-06 Answered
Use a counterexample to show that the statement is false.
T:R2R2,T(x2,x2)=(x1+4,x2) is a linear transformation?
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Expert Answer

oppturf
Answered 2021-01-07 Author has 94 answers
Approach:
Let V and W be vector spaces. The function T:VW is a linear transformation of V into W when the two properties below are true for all u and v in V and for any scalar c.
T(u+v)=T(u)+T(v)
T(cu)=cT(u)
Calculation:
Assume the two vectors u=(u1,u2)andv=(v1,v2).
The sum of the two vectors is,
u+v=(u1,u2)+(v1,v2)
=(u1+v1,u2+v2)
Apply transformation on both side of the above equation.
T(u+v)=T(u1+v1,u2+v2)
=(u1+v1+4,u2+v2)(1)
The sum of the two transformations is,
T(u)+T(v)=T(u1,u2)+T(v1,v2)
=(u1+4,u2)+(v1+4,v2)
=(u1+v1+8,u2+v2)(2)
From equation (1) and equation (2),
T(u+v)T(u)+T(v)
From the above result, the function T(x1,x2)=(x1+4,x2) is not a linear transformation.
Therefore, the statement is false.
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