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.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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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.
Transcribed Image 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.
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