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Resolving Inconsistencies in Linear Systems
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In the linear equation system Ax=b, inconsistencies can arise when the vector b is not within the column space of A. A common solution is to add a column of 1's to matrix A, which expands the column space by introducing a new direction of reachability, allowing previously unreachable vectors like b to be included in the expanded span. This process doesn't rotate the column space but rather introduces a uniform shift, similar to how adding a constant in y=mx+b shifts the line vertically, transforming the linear system into an affine one. This matters because it provides a method to resolve inconsistencies in linear systems, making them more flexible and applicable to a wider range of problems.
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