In P2 is S2 x2 1 x2 x x 6 a linearly independent set Jus

In P_2 is S_2 = {x^2 + 1, x^2 + x, x + 6} a linearly independent set? Justify your answer! Does S_2 span P_2 ? Justify your answer! Is S_2 a basis for P_2? (clearly state why or why not)

Solution

lets calculate the dimension of S_2

It is clear S2 is a subspace of P2 since every element of S2 is in P2.

So if the dimension of S2 is 3 i.e. all elements of S2 are linearly independent then S2 spans P2 and it is a basis for P2

otherwise Not.

Now lets see if all elements are LI

if c1 (x^2 +1) + c2(x^2 + x) + c3(x+6) = 0 (zero polynomial)

then

c1 = -c2

c1 = -6c3

c2 = -c3

which gives all c1,c2,c3 = 0

so they are linearly independent and hence its a basis for P2.