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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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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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.