Justify the following statements 1 If a matrix A is mn and i

Justify the following statements:

1/ If a matrix A is m*n and if the equation Ax=b has a solution for every b, then the columns of A must be linearly independent in R^m.

2/ If A is a 5*5 matrix such that Ax=b has a solution for every b, then the columns of A span R^5.

Solution

1)Since 0 is an element of R m, then by the given assumptions Ax = 0 has at most one solution. We know that Ax = 0 always has the trivial solution, so it follows that Ax = 0 has only the trivial solution. The matrix equation Ax = 0 corresponds to the vector equation x1a1 + · · · + xnan = 0, so this vector equation has only the trivial solution. But, this is exactly the definition of linear independence, so the columns of A must be linearly independent.

2)using above explanation we can say that the columns are linearly independent,therefore they span R5