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 :)
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 $Z$.