Let $f : \mathbb{R}^{3} \to \mathbb{R}^{3}$ be the linear transformation determined by $f(\vec{x}) = A \vec{x}$ where $$A=\left[\begin{array}{ccc} -3 &1 &-3\cr 3 &4 &0\cr -3 &2 &-1 \end{array}\right]$$

1. Find bases for the kernel and image of $f$. Enter a list of vectors, or empty if the basis is the empty set.
   A basis for $\mathrm{Kernel}(f)$ is $\Big\lbrace$ $\Big\rbrace.$
   A basis for $\mathrm{Image}(f)$ is $\Big\lbrace$ $\Big\rbrace.$
   
   
2. The dimension of the kernel of $f$ is and the dimension of the image of $f$ is .
   
   
3. The linear transformation $f$ is
   injective, but not surjective
   surjective, but not injective
   bijective
   none of these