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## Main Question or Discussion Point

Find a basis for Ker T that contains S = [tex]\begin{pmatrix}

1\\

0\\

1\\

0\\

\end{pmatrix}[/tex], [tex]\begin{pmatrix}

0\\

1\\

0\\

2\\

\end{pmatrix}[/tex] where [tex]T : R^4 -> R^4[/tex] is defined by

[tex]T\begin{pmatrix}

1\\

b\\

c\\

d\\

\end{pmatrix} = \begin{pmatrix}

a - b - c\\

a - 2b + c\\

0\\

0\\

\end{pmatrix}[/tex].

Well, I have found a basis 'B' for Ker (T) to be B ={[tex]\begin{pmatrix}

3\\

2\\

1\\

0\\

\end{pmatrix}[/tex], [tex]\begin{pmatrix}

0\\

0\\

0\\

1\\

\end{pmatrix}[/tex]}.

I noticed that the two sets of vectors S and B are linearly independent of one another. Does this mean that there is no basis for Ker(T) that contains S, as a basis must be a minimal spanning set? Or have I gone astray somewhere?

1\\

0\\

1\\

0\\

\end{pmatrix}[/tex], [tex]\begin{pmatrix}

0\\

1\\

0\\

2\\

\end{pmatrix}[/tex] where [tex]T : R^4 -> R^4[/tex] is defined by

[tex]T\begin{pmatrix}

1\\

b\\

c\\

d\\

\end{pmatrix} = \begin{pmatrix}

a - b - c\\

a - 2b + c\\

0\\

0\\

\end{pmatrix}[/tex].

Well, I have found a basis 'B' for Ker (T) to be B ={[tex]\begin{pmatrix}

3\\

2\\

1\\

0\\

\end{pmatrix}[/tex], [tex]\begin{pmatrix}

0\\

0\\

0\\

1\\

\end{pmatrix}[/tex]}.

I noticed that the two sets of vectors S and B are linearly independent of one another. Does this mean that there is no basis for Ker(T) that contains S, as a basis must be a minimal spanning set? Or have I gone astray somewhere?