Determine whether the function is a linear transformation. T: P2 → P2, T(a0 + a1x + a2x2) = (a0 + a1 + a2) + (a1 + a2)x + a2x2.

Question

Determine whether the function is a linear transformation. T: P2 → P2, T(a0 + a1x + a2x2) = (a0 + a1 + a2) + (a1 + a2)x + a2x2.

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Alexandra 1 week 2021-10-12T09:24:40+00:00 1 Answer 0

Answers ( )

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    2021-10-12T09:25:46+00:00

    Answer with Step-by-step explanation:

    We are given that a function

    T:P_2\rightarrow P_2

    T(a_0+a_1x+a_2x^2)=(a_0+a_1+a_2)+(a_1+a_2)x+a_2x^2

    We have to determine the given function is a linear transformation.

    If a function is linear transformation then it satisfied following properties

    1.T(x+y)=T(x)+T(y)

    2.T(ax)=aT(x)

    T(a_0+a_1x+a_2x^2+b_0+b_1x+b_2x^2)=T((a_0+b_0)+(a_1+b_1)x+(a_2+b_2)x^2)=(a_0+b_0+a_1+b_1+a_2+b_2)+(a_1+b_1+a_2+b_2)x+(a_2+b_2)x^2

    T(a_0+a_1x+a_2x^2+b_0+b-1x+b_2x^2)=(a_0+a_1+a_2)+(a_1+a_2)x+a_2x^2+(b_0+b_1+b_2)+(b_1+b_2)x+b_2x^2

    T(a_0+a_1x+a_2x^2+b_0+b-1x+b_2x^2)=T(a_0+a_1x+a_2x^2)+T(b_0+b_1x+b_2x^2)

    T(a(a_0+a_1x+a_2x^2))=T(aa_0+aa_1x+aa_2x^2)

    T(a(a_0+a_1x+a_2x^2))=(aa_0+aa_1+aa_2)+(aa_1+aa_2)x+(aa_2)x^2

    T(a(a_0+a_1x+a_2x^2))=a(a_0+a_1+a_2)+a(a_1+a_2)x+aa_2x^2=a((a_0+a_1+a_2)+(a_1+a_2)x+a_2x^2)=aT(a_0+a_1x+a_2x^2)

    Hence, the function is a linear transformation because it satisfied both properties of linear transformation.

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