Find a minimal spanning set of a set of matrices

Question

I'm supposed to find a minimal spanning set of $W = \{A \in M_n(\mathbb{R}) | \operatorname{Tr}(A) = 0\}$

First of all, what is a minimal spanning set? I can't find the term anywhere in the notes my teacher has posted.

Is it simply $\operatorname{Span}(W) = \{ α \cdot\begin{bmatrix} a && a_2 && a_3 &&... && a_n \\ b_1 && b && b_3 &&... && b_n \\ c_1 && c_2 && c &&... && c_n \\ \vdots && \vdots && \vdots&& \ddots && \vdots \\ e_1 && e_2 && e_3 &&... && e \end{bmatrix} | α \in \mathbb{R},\ a + b +c +... +e= 0\}$ ?

Answer 1

A spanning set in $W$ is a collection of elements in $W$ which spans $W$, i.e. a family $f_1, \dots, f_n \in W$ such that each $x \in W$ can be written as $x = a_1 f_1 + \dots a_n f_n$ for real coefficients $a_i$.

The spanning set is minimal if you can't find a spanning family with fewer vectors. The number of vectors in a minimal spanning set of $W$ is the dimension of $W$.

The dimension of $W$ in your example is $n^2 - 1$ since $M_n(\mathbb{R})$ has dimension $n^2$ and $\operatorname{Tr}(A)=0$ is one equation. Therefore, you need to find $n^2 - 1$ linearly independent elements in $W$.

So consider the following matrices:

• the ones which have one entry outside of the diagonal equal to $1$ and $0$ everywhere else (there are $n(n-1)$ of these);
• the ones which have 1 on the entry $(k, k)$ for $k < n$ and $-1$ for the entry $(n,n)$ and $0$ everywhere else (there are $n - 1$ such matrices).

These are clearly in $W$ and linearly independent. Moreover, it is spanning, because $n(n-1) + n-1 = n^2 - n + n - 1 = n^2 - 1$.