Question

Let P_2 be the vector space of all polynomials of degree 2 or less, and define the transformation T: P_2 to P_3 by: [cont in comments]

Answer 1

\[T(p(x))=2x^2p''(x)+p'(x), \forall p(x)\in P_2\] Prove that this is a linear transformation. ... help? Thanks!

Answer 2

T(ap_1(x)+bp_2(x)) = 2x^2(ap_1(x)+bp_2(x))'' + (ap_1(x)+bp_2(x))'
=2x^2[(ap_1(x))'' + (bp_2(x))''] + (ap_1(x))' + (bp_2(x))'
=2x^2[ap_1(x)'' + bp_2(x)''] + ap_1(x)' + bp_2(x)'
= 2ax^2p_1(x)'' + 2bx^2p_2(x)'' + ap_1(x)' + bp_2(x)'
= a(2x^2p_1(x)'' + p_1(x)') + b(2x^2p_2(x)'' + p_2(x)')
= aT(p_1(x)) + bT(p_2(x))

Answer 3

Is this just showing that it preserves addition and scalar multiplication??

Answer 4

Yes... isn't that the definition of a linear transformation?