Abstract Algebra Problem?

Question

Prove that a polynomial of degree <= 2 under addition is an isomorphism with vector addition in R3:

I don't want the answer, but hints to help me would be GREATLY appreciated. Thanks

Answer 1

A polynomial of degree two is Ax^2 + Bx + C. What's a canonical basis of this space? x^2,x,1.

A vector in three space is (x,y,z). What's a canonical basis of this space? (1,0,0),(0,1,0),(0,0,1).

Compare the bases. You should see a one-to-one match up.

Answer 2

Something like this: When you add the polynomials, the coefficients for each power add. That is, you add the square terms to the square terms, and the linear terms to the linear terms, and the constants to the constants. A vector in R3, is added similarly -- you add each component to the same component in the other vector.

Answer 3

can someone expound on this? i stumbled upon it and am curious. i have a passing knowledge of isomorphism and wonder how you would define the mapping function.