TITLE: For what values of $c \in \mathbb{R}$, $[1, c, c^2]$ is a linear combination of $[1, a, a^2]$ and $[1, b, b^2]$
QUESTION [0 upvotes]: I have found that this only works if $c=1$.
First  I took $c = \sqrt{a^2 + b^2}$  and then solved $a +b = \sqrt{a^2 + b^2}$, which holds true when either $a = 0$ or $b=0$ and put these values in the augmented matrix I would get from the problem statement.

REPLY [2 votes]: If $[1,c,c^2]$ is a linear combination of $[1,a,a^2]$ and $[1,b,b^2]$, then
$$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=0.$$
We may compute the determinant of this matrix to be
$$(a-b)(b-c)(c-a).$$
So, this is zero if and only if $a=b$, $b=c$, or $c=a$. If $c\in\{a,b\}$ then $[1,c,c^2]$ can be written as a linear combination of $[1,a,a^2]$ and $[1,b,b^2]$ with one coefficient $1$ and one coefficient $0$; if $a=b$ then the only linear combination of $[1,a,a^2]$ and $[1,b,b^2]$ with first coordinate $1$ is $[1,a,a^2]$, so we only have a solution in this case if $c=a=b$. So, the condition is exactly $c\in\{a,b\}$.
{"set_name": "stack_exchange", "score": 0, "question_id": 4559919}