\[W\]
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 \[\phi_2(x_1, \; x_2, \; x_3, \; x_4)=-11x_1-4x_2+x_3\]
.