<s>[INST] The dimension of the vector space of all trace-zero symmetric matrices Find the dimension of the vector space of all symmetric matrices of order $n \times n$ (real entries) and trace equal to zero.
 [/INST] $$V_F=\{A\in M_M(F)\:A^T=A , Trace(A=0)\}=$$ forall $A\in V_F$ we have $trac(A)=0$ and $a_{ij}=a_{ji}$ such that $A=[a_{ij}]_{n \times n}$
$$A=\begin{pmatrix}
     a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
     a_{12} & a_{22} & a_2^2 & \cdots & a_2^n \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     a_{1n} & a_{2n} & a_{3n} & \cdots & (-a_{11}-...-a_{n-1 n-1})    
     \end{pmatrix}=a_{11}\begin{pmatrix}
     1 & 0 & 0 & \cdots & 0 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots & -1    
     \end{pmatrix}+a_{22}\begin{pmatrix}
     0 & 0 & 0 & \cdots & 0 \\
     0 & 1 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots & -1    
     \end{pmatrix}+...+a_{n-1n-1}\begin{pmatrix}
     0 & 0 & 0 & \cdots & 0 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots & -1    
     \end{pmatrix}+a_{12}\begin{pmatrix}
     0 & 1 & 0 & \cdots & 0 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots & 0    
     \end{pmatrix}+a_{13}\begin{pmatrix}
    0 & 0 & 1 & \cdots & 0 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots &     
     \end{pmatrix}+....+a_{1n}\begin{pmatrix}
     0 & 0 & 0 & \cdots & 1 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\
     0 & 0 & 0 & \cdots & 0    
     \end{pmatrix}+........+a_{n-1n-1}\begin{pmatrix}
    0 & 0 & 0 & \cdots & 0 \\
     0 & 0 & 0 & \cdots & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1 \\
     0 & 0 & 0 & \cdots & 0    
     \end{pmatrix}\Rightarrow$$$$V_{F}=\{\sum_{i=1}^{n-1}a_{i}E_{ii}+\sum_{i\gt j}c_jE_{ij}\}=<E_{11},E_{22},...,E_{n-1\times n-1},E_{12,....E_{1n},.....,E_{n-1\times n}}>=<X>$$ such that  $ X =\{E_{11},E_{22},...,E_{n-1\times n-1},E_{12},....E_{1n},.....,E_{n-1\times n}\}$ then $X$ generate $V_F $ and easily we can prove $ X$ is linear independent totally X is basis for $V_F$ and $$|x|=\frac{(n-1)(n+2)}{2}$$
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