Let T be a linear transformation from P_{2} text{into} P_{2} such that {T}{left({1}right)}={x},{T}{left({x}right)}={1}+{x}{quadtext{and}quad}{T}{left({x}^{2}right)}={1}+{x}+{x}^{2}. Find {T}{left({2}-{6}{x}+{x}^{2}right)}

glamrockqueen7 2021-01-10 Answered
Let T be a linear transformation from P2 into P2 such that
T(1)=x,T(x)=1+xandT(x2)=1+x+x2.
Find T(26x+x2)
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Expert Answer

lamanocornudaW
Answered 2021-01-11 Author has 85 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 thue for all u and v in V and for any scalar c.
T(u+v)=T(u)+T(v)
T(cu)=cT(u)
Calculation:
The value to be find is T(26x+x2).
26x+x2=c1(1)+c2(x)+c3x2
Compare the coefficient of same power.
c1=2
c2= 6
c3=1
Consider a linear transformation T such that,
v=c1v1+c2v2++cnvn
Then,
T(v)=T(c1v1+c2v2++cnvn)
=c1T(v1)+c2T(v2)++cnT(vn)
So,
T(26x+x2)=2T(1)6T(x)+T(x2)(1)
Subtitute x for T(1),(1+x)forT(x)and(1+x+x2)forT(x2) in equation (1).
T(26x+x2=2(x)6(x+1)+(1+x+x2)
2x6x6+1+x+x2
=53x+x2
Therefore, the value of T(26x+x2)is53x+x2.
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