Suppose $f(\mathbf{e_n}) = \mathbf{a_n}$ for $n = 1, 2$ and $f$ is a linear transformation.

1. $f : \mathbb{R}^k \to \mathbb{R}^n$ for $k=$ and $n=$ .
2. The set of vectors $\lbrace \, \mathbf{e_1}, \mathbf{e_2} \rbrace \,$ is (select all that apply):
   A. linearly independent
   B. a basis for the codomain
   C. a spanning set
   D. a basis for the domain
   E. none of these
   
3. The set of vectors $\lbrace \, f(\mathbf{e_1}), \, f(\mathbf{e_2}) \, \rbrace$ is (select all that apply):
   A. linearly independent
   B. a spanning set
   C. a basis for the domain
   D. a basis for the codomain
   E. none of these
   
4. The linear transformation $f$ is (select all that apply):
   A. surjective (onto)
   B. bijective (an isomorphism)
   C. injective (one-to-one)
   D. none of these
   
5. Using that $f$ is a linear transformation, find $f( 5 \mathbf{e_1} + 2 \mathbf{e_2} )$. Enter your answer as a coordinate vector such as <1,2>.
   $f( 5 \mathbf{e_1} + 2 \mathbf{e_2} ) =$
6. Find the matrix for the linear transformation $f$ (relative to the standard basis in the domain and codomain). That is, find the matrix $A$ such that $f(\mathbf{x}) = A \mathbf{x}$. For instance, enter [ [1,2], [3,4] ] for the matrix ${\left[\begin{array}{cc} 1 &2\cr 3 &4 \end{array}\right]}$.
   $A=$