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I have a homework problem where I need to find the values of a variable within a matrix, A, that result in Nul(A) being spanned by a single vector. My intuition is telling me that the only single vector that can span the null space must be the zero vector, but I'm not 100% certain and would just like to confirm.

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  • $\begingroup$ The zero vector is in the span of any vector. Simply multiply the vector by 0. Another way to see this is to just think about the span of one vector being a vector space on its' own -- which means it would necessarily include the 0 vector. If your null space is spanned by a single vector, this should intuitively tell you that the linear transformation kills exactly one dimension, that its' rank should be 1 less than the dimension of the space, etc. $\endgroup$
    – Square
    Oct 31, 2020 at 14:48
  • $\begingroup$ No, not correct. Think about it some more. Do some computations of null spaces of various $2 \times 2$ matrices. $\endgroup$
    – GEdgar
    Oct 31, 2020 at 14:48
  • $\begingroup$ No, consider the null space of $$\begin{pmatrix} 1 & -1 \\ 1&-1\end{pmatrix}.$$ The null space is spanned by $\binom 1 1$. $\endgroup$
    – User8128
    Oct 31, 2020 at 14:49
  • $\begingroup$ One dimension lower, the null space of the $1x1$ matrix $(0)$ is spanned by $1$-vector $(1)$. $\endgroup$
    – Lee Mosher
    Oct 31, 2020 at 14:59

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Not at all. Take, say, $A=\left[\begin{smallmatrix}1&1\\2&2\end{smallmatrix}\right]$. Then $\operatorname{null}(A)$ is spanned by the vector $(1,-1)$. So, it is spanned by a single vector (which is not the same thing as asserting that there is a single vector that spans it).

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  • $\begingroup$ I see! that makes a lot more sense. Would it be correct to say that a null space is spanned by one vector as long as there is (at most) one free variable? $\endgroup$ Oct 31, 2020 at 15:01
  • $\begingroup$ No. The matrix $\left[\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right]$ doesn't have a pivot in each column, but $\dim\operatorname{null}\left(\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]\right)=1$. $\endgroup$ Oct 31, 2020 at 15:05
  • $\begingroup$ I just read up on ranks and dimensions; I think I've finally narrowed it down: as long as the matrix has 1 or less free variables, its null space will be spanned by a single vector? $\endgroup$ Oct 31, 2020 at 15:08
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    $\begingroup$ Yes, that is correct. $\endgroup$ Oct 31, 2020 at 15:12
  • $\begingroup$ Thank you very much for your help :))) $\endgroup$ Oct 31, 2020 at 15:13

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