Let $\mathbb{A}^m ={\text{Spec }}\mathbb{C}[x_1,\dots,x_m]$, and the multiplicative group $\mathbb G_m \cong \mathbb{C}^*$ acts on $\mathbb{C}[x_1,\dots,x_m]$ by $$\lambda_t (x_1,\dots,x_m)= (t^{a_1}x_1,\dots,t^{a_m}x_m)$$ suppose for all $i, a_i >0$.
Then it is claimed the GIT quotient $\mathbb{A}^m//\mathbb G_{m}$ is empty, but I did not see why this is true. I am not good at GIT stuff, and I don't know why this is different from $$\mathbb{A}^m//G_{m} = \text{Spec }\mathbb{C}[x_1,\dots,x_m]^{G_m} = \text{Spec }\mathbb{C}?$$