## The Vector Space Consisting of All Traceless Diagonal Matrices

## Problem 79

Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.

That is,

\begin{equation*}

V:=\left\{ A=\begin{bmatrix}

a_{11} & 0 & \dots & 0 \\

0 &a_{22} & \dots & 0 \\

0 & 0 & \ddots & \vdots \\

0 & 0 & \dots & a_{nn}

\end{bmatrix} \quad \middle| \quad

\begin{array}{l}

a_{11}, \dots, a_{nn} \in \C,\\

\tr(A)=0 \\

\end{array}

\right\}

\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

**(a)** Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

**(b)** Show that matrices

\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

**(c)** Find the dimension of $V$.

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