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 ?
Relationship between nullspace and row/column space
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linear-algebra
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1**Hint**: The rank is the dimension of the rowspace is the dimension of the columnspace. – 2012-02-08
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1Do you know rank nullity theorem? – 2012-02-08
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0I know that $rank (A) + \text{nullity} (A) = n$, however I do not see how this will help me. I feel so stupid now =( – 2012-02-08
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0What is the dimension of a line through the origin? – 2012-02-08
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0I think I might have gotten this now. The dimension is three right? – 2012-02-08
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0$n$ is $3$ (the matrix is $3\times 3$); the dimension of a *line through the origin* is... – 2012-02-08
1 Answers
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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.