Zero Vector Subspace

Last updated Nov 1, 2022

Let $V$ be a Vector Space over Field $F$. Then the Set ${\mathbf{0}}$ is a Vector Subspace of $V$. It is known as the Zero Vector Subspace of $V$.

Let $\mathbf{a}, \mathbf{b} \in {\mathbf{0}}$. This means $\mathbf{a}, \mathbf{b} = \mathbf{0}$. Let $c \in F$. Then $c \mathbf{a} + \mathbf{b} = c \mathbf{0} + \mathbf{0}= \mathbf{0} \in {\mathbf{0}}$, so ${\mathbf{0}}$ is a Vector Subspace of $V$ since A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition. $\blacksquare$