Basis, a beginner in the field needs some guidance

Question

Show that $B = {p_1(t), p_2(t), p_3(t)}$, where$\\p_1(t)=t^2+t+1 \\p_2(t)=t^2+t+2 \\ p_3(t)=t^2+2t+1$,

is a basis for $P_2$.

What is $(t^2)_B$ $∈ R^3$?

I dont really get the second question here? can someone explain it to me and guide me step by step towards the solution?

Answer 1

You need to show these polynomials are linearly independent.$$\\p_1(t)=t^2+t+1 \\p_2(t)=t^2+t+2 \\ p_3(t)=t^2+2t+1$$

That is

$$ c_1(t^2+t+1) + c_2(t^2+t+2 )+c_3(t^2+2t+1)=0 \implies c_1=c_2=c_3=0$$

Then you have to find coefficients $c_1,c_2,c_3$ such that $$ c_1(t^2+t+1) + c_2(t^2+t+2 )+c_3(t^2+2t+1)=t^2$$

These coefficients are coordinates of $t^2$ with respect to the basis $B =\{p_1,p_2,p_3\}$

Answer 2

Hint: Write the matrix expressing the elements of $B$ with respect to the canonical basis $1,t,t^2$. Prove that the matrix is invertible by computing its determinant for instance.