Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

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##### Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn

##### Equivalence theorem: the columns of A span R^n (or C^n). math.la.t.equiv.col.span

##### Equivalence theorem: there is a pivot position in every row of A. math.la.t.equiv.row.pivot

##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb