Prove or provide counterexample: if the coefficient matrix of a system of linear equations in unknowns has rank , then the system has a solution.
I have an idea for the proof but I am not sure how to formalize it. Some advice, please?
Here's the proof: let be the coefficient matrix in the system . If has rank of , then . Say that's not true - then , implying that the maximum rank of which equivalent to the max number of linearly independent columns, has a maximum of , which is less than by assumption, meaning that the rank will never equal . So we must have . Then if the rank of is , we can reduce the matrix [A|b] using row operations to a form [A'|b'] where all have coefficients of . Then a solution becomes for and for , we have .