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A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?

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There is the all zero solution (i.e. the trivial solution).

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A system is defined as inconsistent if its row-reduced echelon form contains a row of form $\begin{bmatrix} 0 & 0 & 0 & ... & 0 & | & k \end{bmatrix}$ where $k \neq 0$ and | is a separator within augmented matrix. Since your system equals $\vec{0}$, it is impossible to have $k \neq 0$, rendering the system consistent.

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  • $\begingroup$ @Jonas Meyer, I edited the answer -- does that work? I was implicitly talking about rref. $\endgroup$ Jan 13, 2011 at 17:40
  • $\begingroup$ Yes, it is clear, thank you. (I deleted my obsolete question.) $\endgroup$ Jan 13, 2011 at 17:43
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HINT $\ $ Zero is a root of every linear map $\rm\:A\:,\:$ since linear maps must preserve $\rm\ 0 + 0 = 0\:,\ $ i.e.

$$\rm\ A\ (0 + 0\ =\ 0)\ \ \to\ \ A(0) + A(0)\ =\ A(0)\ \ \Rightarrow\ \ A(0) = 0$$

More generally: monoid homomorphisms preserve idempotents, but the only idempotent element in a cancellative monoid is the neutral element $\rm\ a + a = a\ \Rightarrow\ a = 0\:.$

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Because in all the cases,

a) Whether coefficient matrix is singular (infinite number of solution) or non singular (trivial solution). b) Number of rows is less than that of variables.

It has a solution.

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