A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?
Why are all homogenous systems consistent?
4 Answers
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.
There is the all zero solution (i.e. the trivial solution).
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.
-
0Yes, it is clear, thank you. (I deleted my obsolete question.) – 2011-01-13
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 $\rm\ a + a = a\ \Rightarrow\ a = 0\:.$