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The number of linearly independent solution of the homogeneous system of equations $AX=O$, where $X$ consists of $n$ unknowns and $A$ consist of $m$ linearly independent rows can't be equal to $n$

(True/False)

We know that for such a system the number of linearly independent solutions is given by $n-m$. When $n-m = n$ This implies $m=0$. That means number of linearly independent rows is $0$. So rank of $A$ should be $0$. And nullity is $n$. Is it true that if $A$ is a null matrix then there are $n$ linearly independent solution.

It seems like, all $n$ column vectors are solution for this system if $A$ is null matrix. And dimension of all $n$ column vectors is $n$. So the statement should be false. Am I correct$?$

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  • $\begingroup$ For such a system there is either one solution or infinitely many solutions. $\endgroup$ – user247327 Sep 17 '19 at 11:23
  • $\begingroup$ Yes. You have provided an example for which the statement fails; namely, if $A$ has rank zero. $\endgroup$ – Bubaya Sep 17 '19 at 11:24
  • $\begingroup$ @user247327 I know that but this problem asks about possibility of number of linearly independent solutions. $\endgroup$ – Mathaddict Sep 17 '19 at 11:38
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I think you missed an important point :

If $(v_1, \dots, v_n)$ are linearly independent vectors, then $v_i \neq 0$ for all $1\leqslant i \leqslant n$.

If one of them, say $v_1$ is null, take $$\lambda_i = \left\{ \begin{matrix} 0 \textrm{ if } i\neq 0\\ 1 \textrm{ if } i = 1 \end{matrix} \right.$$ Then $\sum_{i=1}^n \lambda_i v_i = 0$ but the $\lambda_i$s are not identicaly $0$.

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  • $\begingroup$ Yes I know that, but how is this point going to help in this problem? $\endgroup$ – Mathaddict Sep 17 '19 at 11:36
  • $\begingroup$ @Mathsaddict Well, if $A$ is null, its column vectors are also null. I can't make sense out of the last paragraph in your question. $\endgroup$ – Olivier Roche Sep 17 '19 at 11:49
  • $\begingroup$ I mean, if $A$ is a null matrix, then all n tuples in $R^{n}$ are solutions of this system. $\endgroup$ – Mathaddict Sep 17 '19 at 11:52

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