which of the following is a basis for P2

a){1-3x+2x(^2) 1+x+x+4x^2;1-7x}
b){1+x+x(^2)x+x^2;x^2}

Question

which of the following is a basis for P2

a){1-3x+2x(^2) 1+x+x+4x^2;1-7x}
b){1+x+x(^2)x+x^2;x^2}

anonymous · Answer

a)$${ 1-3x+2x ^{2}1+x+x+4x ^{2};1-7x}$$
b)$${1+x+x ^{2}x+x ^{2};x ^{2}}$$

anonymous · Answer

Can you think of a very simple basis for P2? (Not one of them)

anonymous · Answer

ya

anonymous · Answer

Actually nevermind that. The two things we need for a basis are: 1) enough vectors to span the vector space and 2) for these vectors to be linearly independent. Can you see how many vectors we will need to span P2?

anonymous · Answer

no

anonymous · Answer

If you represent the polynomial ax^2 + bx + c by the vector (a, b, c) then you will need 3 of them to span P2 (because the dimension of P2 is 3). 
Since both options have 3 polynomials in them, first you need to convert them to vectors (so for example, x^2+x+1 = (1, 1, 1)) and then check that they are linearly independent using matrix operations. Are you able to do that?

anonymous · Answer

why do u choose (1,1,1)?
can u choose any numbers?

anonymous · Answer

I've just converted the polynomial to vector form, the (1, 1, 1) are from the co-efficients of the terms. If we have 3x^2 + 2x+4 it would be (3, 2, 4). Understand?

anonymous · Answer

ok

anonymous · Answer

so that one of mine r in the form of sets ,how am i gonna do that?

anonymous · Answer

Then write the vectors as the rows of a matrix and use matrix operations to check if they are linearly independent. (Since a basis is just a set of vectors which span the vector space and are also linearly independent).

anonymous · Answer

I'm guessing that 1+x+x+4x^2 is written correctly so we can simplify to 4x^2 +2x+1?

For the first option we will have as the matrix:
2, -3, 1
4, 2, 1
0, -7, 1

anonymous · Answer

i don't get wat u did

anonymous · Answer

OK I think I know a simpler way in which you can solve this if you don't get that.
A very simple basis for P2 is x^2, x, 1 right?

anonymous · Answer

ya

anonymous · Answer

i don't get  2,-3,1

anonymous · Answer

Ignore the matrix stuff if you don't get that.If you were able to use a combination of the vectors in the question to make each of x^2, x and 1 then they would be a basis as long as they were linearly independent.
The first thing for you to show is that you can make x^2, x and 1 from one of your options. Do you understand?

anonymous · Answer

ya

anonymous · Answer

Excellent :) Good luck.

anonymous · Answer

thanks