Is the following set linearly independent in the space of polynomials of degree ≤ 3?

Question

Is the following set linearly independent in the space of polynomials of degree $\le3$ with real coefficients? $$ \{x^3+2x^2, -x^2+3x+1, x^3-x^2+2x-1\} $$ I am having trouble with this question and would appreciate help on how to solve it.

Answer 1

We can attack this problem directly. Recall that the definition of linear independence is that if $\sum_{i=1}^n a_i p_i = 0$, with $a_i$ scalars and $p_i$ (in our case) polynomials, then all $a_i=0$. Remember that since we are in the space of polynomials, the $0$ in the sum equation is the zero polynomial. Thus suppose we have $$a_1(x^3+2x^2)+a_2(-x^2+3x+1)+a_3(x^3-x^2+2x-1)=0$$

Since $0$ is the zero polynomial, all coefficients of $x^j$ must equal $0$. Thus we must have the following: $$a_1+a_3=0$$ $$2a_1-a_2-a_3=0$$ $$3a_2+2a_3=0$$ $$a_2-a_3=0$$ You can play around with these a bit to show that the only solution is $$a_1=a_2=a_3=0$$ and thus the polynomials are linearly independent.