Mathematics-Online lexicon: Rank and Solvability of Systems of Linear Equations

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	Mathematics-Online lexicon:
	Rank and Solvability of Systems of Linear Equations

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Let $A \in \mathbb{R}^{m\times n}, b \in \mathbb{R}^m .$

The SLE has a solution $\Longleftrightarrow \mathrm{rank} A = \mathrm{rank} (A,b) .$
The SLE has a unique solution $\Longleftrightarrow \mathrm{rank} A = \mathrm{rank} (A,b) = n .$
The SLE has infinitely many solutions $\Longleftrightarrow \mathrm{rank} A = \mathrm{rank} (A,b) = k < n.$

For the proof observe that the ranks of and of the extended matrix are invariant under elementary operations. Therefore we may suppose that the system is in row - echelon form. There the results follow immediately.

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automatically generated 5/17/2011