Dimensions of matrix I need help starting with these questions. I'm confused on how to find the dimension of the following subespaces of matrices.
Question 1: Let $M_{3,3}$ be the vector space of all 3 × 3 matrices defined over $\mathbb{R}$. Calculate the dimension of the following subspaces of $M_{3,3}$
(a) $W = M_{3,3}.$
(b) $X = \{A ∈ M_{3,3} \ | \ A^T = A \}$ the set of all symmetric matrices.
(c) $Y = \{A ∈ M_{3,3} \ | \ A^T = −A \}$ the set of all skew-symmetric matrices.
(d) $Z = \{A ∈ M_{3,3} \ | \ AB = 0 \}$ where $B$ is the following matrix:
$$
B = \left(
\begin{matrix} 
0 & 1 &0 \\ 
0 &0 &1 \\
0 & 0& 0
\end{matrix}\right)$$
I managed to solve part (a) but for part (b) and (c), would a symmetric matrix be: 
\begin{matrix} 
a & b & c \\ 
b & d  &e \\
c & e & f
\end{matrix}
and since we need to know 6 elements thus the dimension will be 6?
and a skew matrix:
\begin{matrix} 
0 & b &c \\ 
-b &0 &d \\
-c & -d& 0
\end{matrix}
since we only need to know for b,c,d, thus the dimension will be 3?

Please do correct me if I'm wrong in any of my explanation. Thank You :)

 A: For each pair $(i,j)$ with $1\leq i, j\leq3$ let $E_{i,j}$ be the matrix having 1 at position $(i,j)$ (row $i$ and column $j$) and zero elsewhere. So,
$$
E_{i,j}(r,s) = \delta_{i,r}\delta_{j,s} = \begin{cases} 1 & \text{if}\ i=r \ \text{and} \ j=s,\\ 
0 & \text{otherwise}\end{cases}
$$
For (a) you can prove that the set of all possible $E_{i,j}$ matrices is a basis of $M_{3,3}$. 
For (b) and (c) you have the right idea. You only need now a basis.
For (d) try to write in simple equations instead of matrices the condition $AB=0$, that is replace your $B$ matrix in the matrix equation and write then the conditions for a matrix to be in $Z$. Then try to find a basis using matrices $E_{i,j}$ being in $Z$.
A: You seem to have the right intuition for questions (b) and (c). In particular, the dimension of the space of symmetric matrices is 6 (corresponding to the six independent parameters $a, b, c, d, e$ and $f$) and the dimension of the space of skew-symmetric matrices is $3$. You should relate this intuition to the actual definition of dimension by writing down basises for these spaces.
For the last question, you should notice that $AB=0$ just means that the first two columns of $A$ are zero, and apply similar logic.
