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

1. Select all of the true statements (there may be more than one correct answer).
   A. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace$ is a linearly independent set
   B. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \rbrace$ is a linearly independent set
   C. $\lbrace \mathbf{a_1}, \mathbf{a_2} \rbrace$ is a linearly independent set
   D. $\mathrm{span} \lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace = \mathbb{R}^3$
   E. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace$ is a basis for $\mathbb{R}^3$
   F. $\mathrm{span} \lbrace \mathbf{a_1}, \mathbf{a_2} \rbrace = \mathbb{R}^3$
   G. $\lbrace \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4} \rbrace$ is not a basis for $\mathbb{R}^3$
   H. $\mathbf{a_1}$ and $\mathbf{a_2}$ are in the null space of $A$
   
2. If possible, write $\mathbf{a_3}$ as a linear combination of $\mathbf{a_1}$ and $\mathbf{a_2}$; otherwise, enter DNE. Enter a1 for $\mathbf{a_1}$, etc.
   $\mathbf{a_3} =$
3. If possible, write $\mathbf{a_4}$ as a linear combination of $\mathbf{a_1}$, $\mathbf{a_2}$, and $\mathbf{a_3}$; otherwise, enter DNE.
   $\mathbf{a_4} =$
4. The dimension of the column space of $A$ is , and the column space of $A$ is a subspace of ? 1 2 3 4 R R^2 R^3 R^4 .
5. Find a basis for the column space of $A$. Enter your answer as a comma separated list of vectors. If necessary, enter a1 for $\mathbf{a_1}$, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>.
   A basis for the column space of $A$ is $\big\lbrace$ $\big\rbrace$
6. The dimension of the null space of $A$ is , and the null space of $A$ is a subspace of ? 1 2 3 4 R R^2 R^3 R^4 .
7. Find a basis for the null space of $A$. Enter your answer as a comma separated list of vectors of the form <a,b,c> or <a,b,c,d> where a,b,... are numbers.
   A basis for the null space of $A$ is $\big\lbrace$ $\big\rbrace$