Determine whether the following statements are true or false and provide a justification for your response.

  1. If \(A\) is a \(127\times 341\) matrix, then \(\nul(A)\) is a subspace of \(\real^{127}\text{.}\)

  2. If \(\dim~\nul(A) = 0\text{,}\) then the columns of \(A\) are linearly independent.

  3. If \(\col(A) = \real^m\text{,}\) then \(A\) is invertible.

  4. If \(A\) has a pivot in every column, then \(\nul(A) = \real^m\text{.}\)

  5. If \(\col(A) = \real^m\) and \(\nul(A) = \{\zerovec\}\text{,}\) then \(A\) is invertible.