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True or False?

If the homogeneous system $Ax = 0$ has only the trivial solution, then the corresponding non-homogeneous Ax = b is consistent for each b.

My answer is: False. Since all homogeneous systems are consistent, that $Ax = 0$ has only the trivial solution only says that the Matrix $A$ has the form $m\times n$ where $m\ge n.$ However, it is not sufficient to say if $Ax = b$ is consistent for each $b$, since a counter-example would be:

$$ \begin{eqnarray} 5x &+& 3y &=& 1\\ 5x &+& 2y &=& 1\\ 5x &+& y &=& 1 \end{eqnarray} $$ True or False?

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    $\begingroup$ What do you mean by "consistent"? $\endgroup$ Nov 18 '17 at 8:25
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    $\begingroup$ that the linear system has a solution $\endgroup$
    – Tommy Ling
    Nov 18 '17 at 11:14
  • $\begingroup$ But the given example has the solution (x = 1/5, y = 0), right? $\endgroup$
    – Ramanujan
    Sep 19 '18 at 10:24
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    $\begingroup$ Possible duplicate of linear algebra homogeneous system $\endgroup$
    – Ramanujan
    Sep 23 '18 at 10:28
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Your guess is right, but the explanation is not entirely accurate.

First, note that saying $A x = 0$ has only the trivial solution is actually equivalent to saying that the nullspace of $A$ only contains the null vector or, still equivalently, the columns of $A$ are linearly independent.

Now, the equation $A x = b $ can only have a solution if $b$ is in the column space of $A$. So, as you pointed out, even when the columns of $A$ are linearly independent, still $A x = b $ may have no solution. If $A$ is square, though, then of course $A x = b $ always has a solution when its columns are linearly independent, since then its columns span the whole space.

(This does not apply to your example, though, since $(1,1,1)$ is clearly in the column space of $A$.)

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  • $\begingroup$ Can you give a good counter example then? $\endgroup$
    – Ramanujan
    Sep 24 '18 at 12:39

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