Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients, viewed as a vector space over Q, and with the usual rules for vector addition and multiplication by...

Extracted text: Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients, viewed as a vector space over Q, and with the usual rules for vector addition and multiplication by scalars. Let T :V → V be defined by T(ax² +bæ +c) = (a+b+c)x+2(a+b+c). You may assume that T is linear. (a) Find a basis for the null space of T. (b) Find a basis for the range of T. (c) Verify the dimension theorem (rank-nullity theorem) for T.