Basis of the kernel

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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?
 

Answers and Replies

HallsofIvy
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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?
Perhaps you should reread the problem. "Find a basis for the kernel of T that includes <1, 0, 1, 0> and <0, 1, 0, 2>" makes no sense as it is easy to see that those two vectors are NOT in the kernel of T and so cannot be in any basis for that kernel.
 
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Alright, that is what I had thought but just wanted to verify it with someone. Thank you very much!
 
HallsofIvy
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893
For future reference: A problem that says "Find a basis for vector space V that include vectors v1 and v2" , assuming v1 and v2 are in V and are independent, means that you are to find a basis that includes those two vectors and possibly more.
 
Last edited by a moderator:

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