Mathematics-Online course: Linear Algebra-Linear Systems of Equations-Direct Methods-Rank and Solvability of SLE

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	Mathematics-Online course: Linear Algebra - Linear Systems of Equations - Direct Methods
	Rank and Solvability of SLE

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[table of contents][page overview]

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.

automatically generated 4/21/2005