Which one of the following is not a basis for the vector spa

Which one of the following is not a basis for the vector space consisting of polynomials of degree less or equal to 2? (Show your work!) {1, 4 + x, 5 + x^3} {x, x + 2, x^2} {x^2, x^2 + x, x^2 - x) Determine if the following sets of vectors is a basis of R^3: v_1 = (4 1 5), v_2 = (3 -1 7); v_1 = (4 1 5), v_2 = (3 -1 7) v_3 = (2 0 3); v_1 = (4 1 5), v_2 = (3 -1 7) v_3 = (7 0 12).

Solution

5.

A basis for a polynomial vector space P = { a1, a2, a3, ..... a,} is a set of vectors that spans the space and is linearly independent.

1. {1, 4+x, 5+x^2}   

To see whether this is linearly independent ... a₀ (1)+ a₁ (4+x) + a₂ (5+x^2) = 0

only when a₀=0, a₁ =0, a₂=0.

And this also spans all the polynomials of the form -- a₀ (1)+ a₁ (4+x) + a₂ (5+x^2)

Hence this is a basis of vector space for polynomials of degree 2 or less.

2. {x, x+2, x^2}   

This is also linearly independent as for   

a₀(x)+a₁(x+2)+a₂(x^2) =0 to be true only when a₀ = a₁ =a₂=0

also we can represent every polynomial of degree 2 or less using this basis.

So this is also the basis.

3. { x^2, x^2+x, x^2-x}

Using this basis we can not make all polynomials of degree 2 or less as polynomials

with only constant terms can not be represented using this basis. Hence this is not a basis.