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.

Yes, this is indeed a subspace. 
To see that it is a subspace we need to check that it is closed under scalar multiplication and addition. 
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.
