A subspace depending on a parameter

Matrix.
$$A = \begin{pmatrix} 1 & 1 & m \\ 1 & m & 1 \end{pmatrix}.$$

Rank. The two rows are proportional iff $m=1$ (both equal to $(1,1,1)$). So:
- If $m=1$: rank $=1$, $\dim F_1 = 3-1=2$.
- If $m\neq1$: rank $=2$, $\dim F_m = 3-2=1$.

Explicit bases. For $m=1$: $F_1=\{x+y+z=0\}$, a plane, with basis $(-1,1,0),(-1,0,1)$. For $m\neq1$: solve the system. Subtracting gives $(1-m)y+(m-1)z=(1-m)(y-z)=0$, so $y=z$. Then $x+(1+m)y=0$, so $x=-(1+m)y$. Basis $((-(1+m),1,1))$.

Conclusion. $\dim F_m=2$ if $m=1$, $1$ if $m\neq1$.