Review Note

Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a singular matrix, what can we say about vector \( \mathbf{v} \) in the following equation:  \[\mathbf{A}\mathbf{v} = \mathbf{0}\]

There exist a non-zero vector \(\mathbf{v}\) that satisfies the equation, that is because the columns of \(\mathbf{A}\) are linearly dependent. On the contray, if  the columns of \(\mathbf{A}\) are linearly independent, and thus make \(\mathbf{A}\) invertible, only the trivial solution \(\mathbf{v}=\mathbf{0}\) would satisfy the equation.