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Let $A$ be an $m \times n$ matrix. If the nullspace is not trivial, is the linear system $A\mathbf{x}=\mathbf{b}$ inconsistent/has infinite solutions?

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It depends on $\mathbf{b}$. Let $f:\mathbb R^n\longrightarrow\mathbb R^m, \ v\mapsto Av$.

The system $A\mathbf{x}=\mathbf b$ is either inconsistent (if $\mathbf b\not\in\text{Im}(f)$) or it has infinite number of solutions (if $\mathbf b\in\text{Im}(f)$). If $\mathbf v$ is a solution then $\mathbf v+\mathbf n$ is a solution $\forall \ \mathbf n$ in the nullspace of $A$.

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