###### Exercise3

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 $$\real^{127}\text{.}$$

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 position in every column, then $$\nul(A) = \real^m\text{.}$$

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

in-context