Let $A$ be a $3\times2$ matrix. Explain why the equation $Ax=b$ cannot be consistent for all $b$ in $\mathbb{R}^3$. Generalize your argument to the case of an arbitrary $A$ with more rows than columns.
Explain why the equation $Ax=b$ cannot be consistent for all $b$ in $\mathbb{R}^3$.
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linear-algebra
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2This looks like homework, what have you tried ? – 2012-09-05
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0Not so much homework as studying for a test. I've been a bit stuck on this one. – 2012-09-05
2 Answers
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HINT: The equation $Ax=b$ is consistent if and only if $b$ is a linear combination of the columns of $A$, i.e., if and only if $b$ is in the column space of $A$. What is the maximum possible dimension of the column space of $A$? What is the dimension of $\Bbb R^3$? (If you get the point of this hint, the generalization should be pretty obvious.)
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0It seems that the maximum possible dimension of A is cartesian space (2). The dimension of R3 is 3D space (3). Therefore, since the dimensions are not equal, I would assume that there is no way that Ax=b could be consistent for all b. – 2012-09-05
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0@Faeynrir: That’s right. The column space of $A$ is a $2$-dimensional subspace of $\Bbb R^3$; geometrically speaking, it’s a plane in $\Bbb R^3$, so it can’t possibly be all of $\Bbb R^3$. – 2012-09-05
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0I think this would imply that b is a plane and Ax is simply a line? Or just within a subspace? Edit: Just reread your comment, I see – 2012-09-05
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0@Faeynrir: No, $b$ is a single point in $\Bbb R^3$. The set of all possible values of $Ax$, as $x$ ranges over $\Bbb R^2$, is a plane in $\Bbb R^3$. – 2012-09-05
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0Great. Thank you very much for your help so far, that makes perfect sense. – 2012-09-05
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I view this as a system of equations. $Ax = b$ gives you $3$ linear equations with $2$ variables.