I have to determine whether the polynomials $p_1(x)=2-x^2$, $p_2(x)=3x$, $p_3(x)= x^2 +x-2$ are linearly dependent or independent but I am not sure how to start. Anyone care to enlighten me?
Also I have to find out if it spans $P^{(2)}$.
[Math] Linear dependency of polynomials question
linear algebra
Best Answer
You need to know whether there are scalars $c_1$, $c_2$, $c_3$ not all $0$ such that $c_1p_1+c_2p_2+c_3p_3=0$. So: $$ \begin{align} & c_1p_1(x)+c_2p_2(x)+c_3p_3(x) \\[6pt] = {} & c_1 (2-x^2)+c_2(3x)+c_3(x^2+x-2) \\[6pt] = {} & (c_3-c_1)x^2 + (3c_2 + c_3)x + 2(c_1-c_3) \\[6pt] = {} & 0 \text{ for all }x. \end{align} $$ That means you need $$ \begin{align} c_3-c_1& =0 \\[6pt] 3c_2 + c_3 & =0 \\[6pt] 2(c_1-c_3) & =0 \end{align} $$ The question is whether than can happen if at least one of the numbers $c_1,c_2,c_3$ is not $0$. And the answer is "yes", as you should be able to figure out from there. Therefore they are linearly dependent.
Three members of a $3$-dimensional vector space cannot span the space unless they're linearly independent. (More than three can, in some cases.)