Determine If the Vector U Is in the Column Space

Multiple Choice

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
-Let $\mathbf { v } _ { \mathbf { 1 } } = \left[ \begin{array} { l } - 4 \\ - 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 3 } } = \left[ \begin{array} { r } 1 \\ - 5 \\ 2 \end{array} \right]$ , and $\mathrm { H } = \operatorname { Span } \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ .
Note that $\mathbf { v } _ { 3 } = 2 \mathbf { v } _ { 1 } - 3 \mathbf { v } _ { 2 }$ . Which of the following sets form a basis for the subspace $H$ , i.e., which sets form an efficient spanning set containing no unnecessary vectors?
$A : \left\{ v _ { 1 } , v _ { 2 } , v _ { 3 } \right\}$
$B : \left\{ v _ { 1 } , v _ { 2 } \right\}$
$C : \left\{ v _ { 1 } , v _ { 3 } \right\}$
D: $\left\{ v _ { 2 } , v _ { 3 } \right\}$

A) B only
B) A only
C) B, C, and D
D) $B$ and $C$