A Basis for the Annihilator of a Subspace

Let  
\[W\]
  be a subspace of  
\[\mathbb{R}^4\]
  spanned by  
\[\left\{ \begin{pmatrix}-1\\2\\-3\\4\end{pmatrix} , \begin{pmatrix}0\\1\\4\\-1\end{pmatrix} \right\}\]
.
The annihilator of  
\[W\]
  is the set of linear functions that sends all elements of  
\[W\]
  to zero.
\[\phi ) \mathbf{w} = \mathbf{0}\]
  for all  
\[\mathbf{w} \in W\]
. Let  
\[\phi \begin{pmatrix}w_1\\w_2\\w_3\\w_4\end{pmatrix}=a_1w_1+a_2w_2+a_3w+3+a_4w_40\]
  then
\[\phi \begin{pmatrix}-1\\2\\-3\\4\end{pmatrix}=-a_1+2a_2-3a_3+4a_4=0\]
  (1)
\[\phi \begin{pmatrix}0\\1\\4\\-1\end{pmatrix}=a_2+4a_3-a_4=0\]
  (2)
(1)-2(2) gives
\[\phi \begin{pmatrix}-1\\2\\-3\\4\end{pmatrix}=-a_1-11a_3+6a_4=0\]
  (3)
\[\phi \begin{pmatrix}0\\1\\4\\-1\end{pmatrix}=a_2+4a_3-a_4=0\]
  (4)
There are 2 equations we 4 unknowns, so we can take 4-2=2 unknowns as arbitrary.
Let  
\[a_3=0, \; a_4=1\]
  then  
\[a_1=6, \; a_2=1, \; a_4=1\]
  and we can take  
\[\phi_1(x_1, \; x_2, \; x_3, \; x_4)=6x_1+x_2+x_4\]
.
Let  
\[a_3=1, \; a_4=0\]
  then  
\[a_1=-11, \; a_2=-4, \; a_3=1 \]
  and we can take  
\[\phi_2(x_1, \; x_2, \; x_3, \; x_4)=-11x_1-4x_2+x_3\]
.

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