Let P3 be the space of all polynomials of degree at most 3 a

Let P_3 be the space of all polynomials of degree at most 3, and B = {1, t, t^2, t^3) the standard basis for P_3. Consider the mappings T and S from P_3 into P_3 defined as follows. T(a_0 + a_1t + a_2t^2 + a_3t^3) = 2a_2 + 3a_1t - 5a_0t^2 + 6a_3t^3 and S(a_0 + a_1t + a_2t^2 + a_3t^3) = 2(a_2 + a_1) + a_0t^2 + (2a_3 + 4)t^3. Is S a linear transformation? If YES, find the B-matrix for S. If NO, explain why.

Solution

Let   P₁ = a+bt+ct²+dt³ and P₂ = m+nt+qt+rt³ be 2 arbitrary elements of P₃ and let be an arbitrary scalar.

Then S (P₁+P₂) = S[ (a+bt+ct²+dt³)+( m+nt+qt+rt³ ) ]= S[ (a+m) +(b+n)t +(c+q)t² +(d+r)t³] = 2( c+q+b+n) + (a+m)t² + [2(d+r)+4]t³ = 2(b+c+n+q)+ (a+m)t² + [2(d+r)+4]t³.

Also, S(P₁) = 2(c+b) + at² +(2d+4)t³ and S(P₂) = 2(q+n) + mt² +(2r+4)t³ so that S(P₁) + S(P₂) = 2(c+b) + at² +(2d+4)t³+2(q+n) + mt² +(2r+4)t³ = 2( b+c+n+q) + (a+m)t²+[ 2(d+r) +8]t³.

Thus, S (P₁+P₂) S(P₁) + S(P₂). Hence S is not a linear transformation as it does not preserve vector addition.