# Determine if S is a subspace of the F-vector space V

$\mathbb{F} = \mathbb{C}$, $V = \mathbb{C}^{3\times 3}$, the set of all complex $3\times 3$ matrices, and S is all the set of all matrices of the form
$$\left( \begin{array}{ccc} a & a & a \\ 0 & 0 & a \\ a & a & a \end{array} \right)$$

where a is an arbitrary complex number.

determine if $S \subset \mathbb{C}^{3\times 3}$ is a subspace.

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Choose any scalar $z\in \mathbb{C}$ then $$z \left( \begin{array}{ccc} a & a & a \\ 0 & 0 & a \\ a & a & a \end{array} \right) = \left( \begin{array}{ccc} za & za & za \\ 0 & 0 & za \\ za & za & za \end{array} \right) \in S$$
and for any $a, b\in \mathbb{C}$ we have, $$\left( \begin{array}{ccc} a & a & a \\ 0 & 0 & a \\ a & a & a \end{array}\right) + \left( \begin{array}{ccc} b & b & b \\ 0 & 0 & b \\ b & b & b \end{array} \right) = \left( \begin{array}{ccc} a+b & a+b & a+b \\ 0 & 0 & a+b \\ a+b & a+b & a+b \end{array} \right) \in S$$
Thus we see that $S$ is closed under scalar multiplication and addition of vectors, so it is a subspace.