Solving the System Ax = b: Is Full Rank Necessary?

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The discussion centers on whether the system Ax = b has a solution when A is an m x n matrix with m > n and has full rank. It is asserted that the statement is false because vector b does not necessarily lie within the range of A, which would mean Ax = b has no solution. Participants suggest that a counterexample is needed, specifically a simple matrix A composed of standard basis elements in R^m. The challenge lies in constructing a valid example that demonstrates this scenario. Ultimately, the conclusion is that full rank is not a sufficient condition for the existence of a solution to Ax = b.
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Homework Statement


The system Ax=b, with Amxn, and m>n, always has a solution when A has full rank. If False, give a counter example, if True, say why.



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The Attempt at a Solution


I want to say False because b doesn't need to be in the range of A, so Ax=b wouldn't have a solution. However, I'm having trouble making a counter example (i.e. A = 3x2 matrix and b = 3x1 matrix) that proves the point.
 
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Your reasoning is correct. For a counterexample, try a very simple A whose columns are elements of the standard basis for \mathbb{R}^m.
 

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