Suppose $\mathbf{a_1}$, $\mathbf{a_2}$, $\mathbf{a_3}$, $\mathbf{a_4}$, and $\mathbf{a_5}$ are vectors in $\mathbb{R}^3$, $A = (\mathbf{a_1} \mid \mathbf{a_2} \mid \mathbf{a_3} \mid \mathbf{a_4} \mid \mathbf{a_5} )$, and

1. Select all of the true statements (there may be more than one correct answer).
   A. $\lbrace \mathbf{a_1}, \mathbf{a_2} \rbrace$ is a basis for $\mathbb{R}^3$
   B. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \rbrace$ is a linearly independent set
   C. $\mathrm{span} \lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \rbrace = \mathbb{R}^3$
   D. $\mathrm{span} \lbrace \mathbf{a_1}, \mathbf{a_2} \rbrace = \mathbb{R}^3$
   E. $\mathrm{span} \lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace = \mathbb{R}^3$
   F. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \rbrace$ is a basis for $\mathbb{R}^3$
   G. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace$ is a basis for $\mathbb{R}^3$
   H. $\lbrace \mathbf{a_1}, \mathbf{a_2} \rbrace$ is a linearly independent set
   I. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace$ is a linearly independent set
   
2. If possible, write $\mathbf{a_4}$ as a linear combination of $\mathbf{a_1}$, $\mathbf{a_2}$, and $\mathbf{a_3}$. (Enter a1 for $\mathbf{a_1}$, a2 for $\mathbf{a_2}$, a3 for $\mathbf{a_3}$.) If $\mathbf{a_4}$ is not a linear combination of $\mathbf{a_1,a_2,a_3}$ enter DNE.
   $\mathbf{a_4} =$
3. The dimension of the column space of $A$ is , and the column space of $A$ is a subspace of .
4. The row reduced echelon form of A reveals a particular set of vectors that forms a basis for the column space of A. List the vectors in that basis. (Of course the column space has many bases, but the rref form suggests a particular basis.) Enter your answer as a comma separated list of vectors.
   The basis for the column space of $A$ that’s suggested by the rref form is $\big\lbrace$ $\big\rbrace$
5. The dimension of the null space of $A$ is , and the null space of $A$ is a subspace of .
6. If $\mathbf{x_1} = {\left<-5,-2,-1,1,0\right>}$, then $A \mathbf{x_1} =$ . Is $\mathbf{x_1}$ in the null space of $A$? choose yes no
7. If $\mathbf{x_2} = {\left<3,-2,1,0,1\right>}$, then $A \mathbf{x_2} =$ . Is $\mathbf{x_2}$ in the null space of $A$? choose yes no
8. If $\mathbf{x_3} = 3 \mathbf{x_2} - 4 \mathbf{x_1} =$ , then $A \mathbf{x_3} =$ . Is $\mathbf{x_3}$ in the null space of $A$? choose yes no
9. Find a basis for the null space of of $A$. Enter your answer as a comma separated list of vectors.
   A basis for the null space of $A$ is $\big\lbrace$ $\big\rbrace$