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Given is a nonhomogeneous system of 7 linear equations with 10 unknowns. It is not possible to change the column vector of constant terms to make the system inconsistent.

Claim: The general solution of the system has 3 free variables.

Prove: Because there are 10 unknowns and 7 linear equations I would conclude you would at least have 3 free variables. I am unsure however if the claim is valid because to me it seems that some of the seven linear equations could also be linearly dependent thus making the claim false. It should be: "has 3 or more free variables".

Thus can a non-homogeneous have linearly dependent equations?

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  • $\begingroup$ The claim talks about the general solution. Choosing vectors randomly, the probability that they are linearly independent is $1$. $\endgroup$ – Qi Zhu Jul 7 '19 at 10:34
  • $\begingroup$ Is the question whether for every $7\times 10$ - matrix $A$ , we can find a vector $v$ with $10$ entries such that $Ax=v$ has no solution ? If so, this depends on the rank of the matrix $A$. $\endgroup$ – Peter Jul 7 '19 at 11:44
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The hypothesis of your linear problem can be rewritten

Let $A\in M_{7,10}(K)$; for every $b\in K^7$, there is $x\in K^{10}$ s.t. $Ax=b$.

That is equivalent to say that $A$ is surjective, that is $rank(A)=7$, that is, $dim(\ker(A))=10-7=3$.

If $x_0$ is a particular solution of $Ax=b$, then its general solution is $x_0+\ker(A)$, an expression that depends on $3$ arbitrary constants -when you choose a basis of $\ker(A)$-.

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