Let A be an mn matrix Prove or disprove the following statem

Let A be an m×n matrix. Prove or disprove the following statement: The equation Ax = b has a solution for all b R m if and only if the equation AT x = 0 has only the trivial (i.e. zero) solution. (Here AT denotes the transposed of A.)

Solution

Either the system of equations have zero solution, one solution or infinite number of solution since

Ax = b has a solution therefore multiplying the sides by A(inv) we get

x = bA(inv) substituting the values in the section equation we get

AT (bA (inv)) = 0 , will have only one real solution since reduced matrix column will have one zero

But this statement will only hold for trivial solution, hence the answer is TRUE