Let L : P2&nbsp;→&nbsp;P3 be a linear transformation for which weknow that L(1) = 1, L(t) = t 2, L(t 2) = t 3 + t.(a) Find L(2t 2 - 5t + 3). (b) Find L(at 2 + bt + c).

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Let L : P2&amp;nbsp;→&amp;nbsp;P3 be a linear transformation for which weknow that L(1) = 1, L(t) = t 2, L(t 2) = t 3 + t.(a) Find L(2t 2 - 5t + 3). (b) Find L(at 2 + bt + c).

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Given:Let L: P2→P3 be a linear transformation for which L(1)=1,L(t)=t2 &amp;amp; L(t2)=t3+tTo determine:a) L(2t2−5t+3) b) L(at2+bt+c)SInce, L: P2→P3 is a linear transformation, therefore, following properties are followed:1) L(u+ν)=L(u)+L(ν) where u, v are the vectors.2) L(au)=aL(u) where, aϵFPart a)Consider a) L(2t2−5t+3) Using above properties, we have,L(2t2−5t+3) =L(2t2)+L(−5t)+L(3)=2L(t2)−5L(t)+3L(1)=2(t3+t)−5(t2)+3(1)=2t3+2t−5t2+3=2t3−5t2+2t+3Hence, L(2t2−5t+3)=2t3−5t2+2t+3Part b)Consider  L(at2+bt+c)Using above properties, we have,L(at2+bt+c)=L(at2)+L(bt)+L(c)=aL(t2)+bL(t)+cL(1)=a(t3+t)+b(t2)+c(1)=at3+at+bt2+c=at3+bt2+at+cHence,  L(at2+bt+c)=at3+bt2+at+c

Let L : P2 \rightarrow P3 be a

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