Independence depending on a parameter

Determinant.
$$\Delta = \begin{vmatrix} 1 & 1 & m \\ 1 & m & 1 \\ m & 1 & 1 \end{vmatrix}.$$
Adding all columns to the first:
$$\Delta = (m+2)\begin{vmatrix} 1 & 1 & m \\ 1 & m & 1 \\ 1 & 1 & 1 \end{vmatrix}.$$
Subtracting the third row from the first two:
$$\Delta = (m+2)\begin{vmatrix} 0 & 0 & m-1 \\ 0 & m-1 & 0 \\ 1 & 1 & 1 \end{vmatrix} = (m+2)\cdot(-(m-1)^2).$$
So $\Delta = -(m+2)(m-1)^2$.

Conclusion. The family is linearly independent if and only if $m\neq -2$ and $m\neq 1$.