Find Basis for Ker (T) with S = {1, 0, 1, 0}

[Luke1294]

Find a basis for Ker T that contains S = $$\begin{pmatrix}<br /> 1\\<br /> 0\\<br /> 1\\<br /> 0\\<br /> \end{pmatrix}$$, $$\begin{pmatrix}<br /> 0\\<br /> 1\\<br /> 0\\<br /> 2\\<br /> \end{pmatrix}$$ where $$T : R^4 -> R^4$$ is defined by

$$T\begin{pmatrix}<br /> 1\\<br /> b\\<br /> c\\<br /> d\\<br /> \end{pmatrix} = \begin{pmatrix}<br /> a - b - c\\<br /> a - 2b + c\\<br /> 0\\<br /> 0\\<br /> \end{pmatrix}$$.

Well, I have found a basis 'B' for Ker (T) to be B ={$$\begin{pmatrix}<br /> 3\\<br /> 2\\<br /> 1\\<br /> 0\\<br /> \end{pmatrix}$$, $$\begin{pmatrix}<br /> 0\\<br /> 0\\<br /> 0\\<br /> 1\\<br /> \end{pmatrix}$$}.

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?

 

[HallsofIvy]

Find a basis for Ker T that contains S = $$\begin{pmatrix}<br /> 1\\<br /> 0\\<br /> 1\\<br /> 0\\<br /> \end{pmatrix}$$, $$\begin{pmatrix}<br /> 0\\<br /> 1\\<br /> 0\\<br /> 2\\<br /> \end{pmatrix}$$ where $$T : R^4 -> R^4$$ is defined by

$$T\begin{pmatrix}<br /> 1\\<br /> b\\<br /> c\\<br /> d\\<br /> \end{pmatrix} = \begin{pmatrix}<br /> a - b - c\\<br /> a - 2b + c\\<br /> 0\\<br /> 0\\<br /> \end{pmatrix}$$.

Well, I have found a basis 'B' for Ker (T) to be B ={$$\begin{pmatrix}<br /> 3\\<br /> 2\\<br /> 1\\<br /> 0\\<br /> \end{pmatrix}$$, $$\begin{pmatrix}<br /> 0\\<br /> 0\\<br /> 0\\<br /> 1\\<br /> \end{pmatrix}$$}.

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.

 

[Luke1294]

Alright, that is what I had thought but just wanted to verify it with someone. Thank you very much!

 

[HallsofIvy]

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.