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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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.