Question about the dimension $S$

Question

let $V$ be the space of real $3\times3$ matrices and let $S$ belongs to $V$ and be the subspace of symmetric matrix. Find $\dim(S)$.

Answer 1

Hint: Show that the following matrices make a basis for $S$ : $$E_1:=\begin{pmatrix} 1 & 0 &0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}; E_2:=\begin{pmatrix} 0 & 1 &0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix};E_3:=\begin{pmatrix} 0 & 0 &1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}\\E_4:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix};E_5:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix};E_6:=\begin{pmatrix} 0 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} $$

Answer 2

Step 1

There are $9$ entries in a $3\times3$ matrix. Because it is symmetric, the entries are reflected along the diagonal.

Question : What is the maximum number of unique entries? (try writing $3\times3$ symmetric matrices and counting)

Step 2

A standard basis vector of the vector space of $3\times3$ matrices is one that has a $1$ in exactly one entry, and a $0$ in all other entries.

Question : If you take the standard basis vector with a $1$ in entry $(i,j)$ and let it also have a $1$ in entry $(j,i)$. What kind of subspace does this new vector span?

Step 3

Try to combine the previous two steps into a basis for the required subspace.

Question: If there are $n$ vectors in a basis for a subspace, what is the dimension of the subspace? What is the dimension of your required subspace?