Exercise 3

Determine whether the following statements are true or false and provide a justification for your response. Unless otherwise stated, assume that $A$ is an $m\times n$ matrix.

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

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

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

4. If $A$ has a pivot position in every column, then $\nul(A) = \real^n$.

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