Show that $(1,1,1)$, $(a,b,c)$, $(a^2, b^2, c^2)$ are linearly indepdenent for distinct $a,b,c$

Note: the three vectors are not linearly independent if $a = b$, or if $a = c$, or if $b = c$. So you will need to assume $a, b, c$ are distinct.

Hint: Define the quadratic polynomial $f(x) = c_1 + c_2 x + c_3 x^2$. What are $f(a), f(b), f(c)$? What can you conclude about the roots of $f$?

You have a linear system of three equations in three unknowns, which you can write in matrix form as $$ A\mathbf{c} = \mathbf{0}, $$ where $$ A = \begin{pmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{pmatrix} \in \mathbb{R}^{3 \times 3} $$ is your coefficient matrix and $$ \mathbf{c} = \begin{pmatrix}c_1\\c_2\\c_3\end{pmatrix} \in \mathbb{R}^3 $$ is your unknown vector. In these terms, then, $(1,1,1)$, $(a,b,c)$, and $(a^2,b^2,c^2)$ are linearly independent if and only if the system $A\mathbf{c} = \mathbf{0}$ has a unique solution, if and only if $A$ is invertible, if and only if $\det(A) \neq 0$. But now, $$ \det(A) = \begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix} = \begin{vmatrix}1&a&a^2\\0&b-a&b^2-a^2\\0&c-a&c^2-a^2 \end{vmatrix}=(b-a)(c^2-a^2)-(b^2-a^2)(c-a)\\=(b-a)(c-a)\left((c+a)-(b+a) \right) = (b-a)(c-a)(c-b), $$ from which it follows that $(1,1,1)$, $(a,b,c)$, and $(a^2,b^2,c^2)$ are linearly independent if and only if $a$, $b$, and $c$ are all distinct.