$T$ is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ such that $T(u) = u$, $T(v) = 2v$, $T(w) = 3w$, where $u,v,w$ are non-zero. Then which are these are necessarily true:
- $det(T) = 6$.
- ${u,v,w}$ is a basis of $\mathbb{R}^3$.
- $T$'s characteristic polynomial is $(x-1)(x-2)(x-3)$.
Or is there not enough information?
Attempt
If ${u,v,w}$ are the standard basis of $\mathbb{R}^3$, then all three results follow. But I am kinda stuck when considering ${u,v,w}$ that are not the standard basis.