Linearly independence over C and over R Given $\{ix^2 +x -i, x^2-ix+1,x^2 -ix\}$ , Is it necessary linearly independent over C? over R?
Well, I did prove that it's necessary independent over C and over R but it's obviously wrong, I need your help to prove it right.
So I put it on a matrix like that:
$$
        \begin{matrix}
        i & 1 & -1  \\
        1 & -i & 1  \\
        1 & -i & 0  \\
        \end{matrix}
$$
And then I  got
$$
        \begin{matrix}
        1 & -i & 1  \\
        0 & -1 & -i-1  \\
        0 & 0 & -1  \\
        \end{matrix}
$$
And I explained that because they have complex variables inside the matrix and it's linearly independent then it's linearly independent over C, and because $ R \subset C $ then it's also true over R but I don't think that I can say that which means it won't be necessary independent over R at all. What do you guys think?
 A: In this problem you are given a collection
$$
\beta=\{ix^2 +x -i, x^2-ix+1,x^2 -ix\}
$$
of polynomials in $\Bbb C[x]$. Note that $\Bbb C[x]$ can be considered as both an $\Bbb R$-vector space and a $\Bbb C$-vector space. You are asked if $\beta$ is independent over $\Bbb R$ or $\Bbb C$.
To see if $\beta$ is independent over a field $F$, suppose
$$
\lambda_1\cdot(ix^2 +x -i)+\lambda_2\cdot(x^2-ix+1)+\lambda_3\cdot(x^2 -ix)=\mathbf{0}\tag{1}
$$
where $\lambda_1,\lambda_2,\lambda_3\in F$ are not all zero. Note that (1) is equivalent to
$$
\begin{bmatrix}
i & 1 & 1 \\
1 & -i & -i \\
-i & 1 & 0
\end{bmatrix}
\begin{bmatrix}\lambda_1\\ \lambda_2\\ \lambda_3\end{bmatrix}=
\begin{bmatrix}0\\0\\0\end{bmatrix}\tag{2}
$$
But 
$$
\DeclareMathOperator{Null}{Null}\Null
\begin{bmatrix}
i & 1 & 1 \\
1 & -i & -i \\
-i & 1 & 0
\end{bmatrix}=
\DeclareMathOperator{Span}{Span}\Span_{\Bbb C}\left\{
\begin{bmatrix}
i\\ -1\\ 2
\end{bmatrix}
\right\}
$$
so $\begin{bmatrix}\lambda_1\\ \lambda_2\\ \lambda_3\end{bmatrix}$ is a solution to (2) if and only if
$$
\begin{bmatrix}\lambda_1\\ \lambda_2\\ \lambda_3\end{bmatrix}
=
\lambda\cdot
\begin{bmatrix}
i\\ -1\\ 2
\end{bmatrix}
$$
for some $\lambda\in\Bbb C$. This is possible if $F=\Bbb C$ but not possible if $F=\Bbb R$ (can you prove this?). Hence $\beta$ is independent over $\Bbb R$ but not independent over $\Bbb C$.
