prove that P2, the vector space of all real polynomials of degree at most 2, is
ID: 1892237 • Letter: P
Question
prove that P2, the vector space of all real polynomials of degree at most 2, is isomorphic to R3, the vector space of all real column vectors of length 3Explanation / Answer
Define f : R3 -> P2 by f(a,b,c) = a + bx + cx^2 We must show f is bijective and homomorphic. Assume f(a,b,c) = f(d,e,f). Then a + bx + cx^2 = d + ex + fx^2 => (a-d) + (b-e)x + (c-f)x^2 = 0 Thus a-d = 0 => a = d b-e = 0 => b = e c-f = 0 => c = f Hence (a,b,c) = (d,e,f) and f is injective. Let z be in P2. Then z = m + nx + px^2 for some m,n,p. Thus f(m,n,p) = z. Therefore f is surjective (and thus bijective). Now consider f((a,b,c)+(d,e,f)) = f(a+d,b+e,c+f) = (a+d) + (b+e)x + (c+f)x^2 = (a + bx + cx^2) + (d + ex + fx^2) = f(a,b,c) + f(d,e,f) Therefore f is a homomorphism. So we have shown f is a bijection and a homomorphism. Therefore the two spaces are isomorphic :D
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