Question

Let p5 be the set of polynomials of degree 5 at most. Find the dimension of p5. Find two bases for p5 that have no vectors in common. Prove that your choices are bases.

Answer 1

i think that should be{1, x, x^2, x^3, x^4, x^5} n dim(P5) =6. well, i didn't get this_"Find two bases for p5 that have no vectors in common". if v1,...,vn and w1,...wm are bases for P5,then m=n. so what are the actually 2 bases for P5??? can i say v1,...,v5 and w1,...,w5 ???

Answer 2

Yes, the dimension is 6, and the basis you provided is a basis. It's essentially the standard basis for that space. For another basis.... multiply each element of the basis by 2, voila, new basis :P

Answer 3

Remember, any independent set of vectors that spans the space is a basis. There are lots of bases.

Answer 4

oh yes, so (2, 3x, 4x^2, 5x^3, 6x^4, 7x^5) is a basis as well?

Answer 5

Yep. It's independent and it spans the space so there you go.

Answer 6

cool' thank you very much!