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Prove that T : R 3 → P2 defined by T(a, b, c) = ax^2 +...

Prove that T : R 3 → P2 defined by T(a, b, c) = ax^2 + bx + c yields a vector space isomorphism

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Answer #1

first we find matrix representation A of T.then row reduced echelon form(RREF A) of T is A itself.then rank A=3(number of nonzero rows in RREFA) implies T is onto.

Solving (RREFA X)=0 we get nullspace A.since nullspace is trivial, nullity T=0.so by a theorem T is oneone

So T is one one and onto implies T is an isomorphism.

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