For Problems 914 determine whether the given set S of vector

For Problems 9-14, determine whether the given set S of vectors is a basis for Pn (R).

Solution

Here the vector space is V=P₂(R)=set of all polynomials of degree atmost 2 over the field of real numbers.

We know tha {1, x, x²} is the standard basis of this vector space and hence dim(V)=dim(P₂(R)) =3

Now in the given set S={-2x+x^2, 1+2x+3x² , -1-x² , 5x+5x² } there are FOUR elements. A set \'S\' of FOUR elements can not be independent and hence cannot be a basis for a vectrospace of dimension THREE.

Therefore given set S is not a basis for the vecor space P₂(R)