# Relationship between nullspace and row/column space

Suppose I have a $3\times 3$ matrix $A$, whose null space is a line through the origin in $3$-space. Can the row or column space of $A$ also be a line through the origin ?

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Hint: The rank is the dimension of the rowspace is the dimension of the columnspace. – Arturo Magidin Feb 8 '12 at 21:29
Do you know rank nullity theorem? – Davide Giraudo Feb 8 '12 at 21:30
I know that $rank (A) + \text{nullity} (A) = n$, however I do not see how this will help me. I feel so stupid now =( – N3buchadnezzar Feb 8 '12 at 21:32
What is the dimension of a line through the origin? – Davide Giraudo Feb 8 '12 at 21:35
I think I might have gotten this now. The dimension is three right? – N3buchadnezzar Feb 8 '12 at 21:42

Since the null-space of $A$ is a line, which is a 1-dimensional subspace, the rank-nullity theorem tells us, that the rank of the matrix, which is the dimension of its row/column-space, is 2 and therefore the column-space cannot be a line, but a plane, a 2-dimensional subspace.