linear span of subspace

Question

we have the following subspace over $\mathbb{R}$ $$M = \{ A \in M^{{n\times n}} | A = -\overline{A} \}$$ I found that it is a subspae and now I need to find the linear span of it. how can I calculate the linear span of a subspace?

Answer 1

If a subset is already a subspace, then it equals it own span (which is defined as the smallest supspace containing it)

Answer 2

First note that for every matrix $A=(a_{j,k})$ satisfying $A = -\bar{A}$, gives us, $a_{j,k}= -\bar{a_{j,k}}$ i.e. $a_{j,k}$ is purely imaginary. So $a_{j,k}= i b_{j,k}$, where $b_{j,k}$ is a real no. Now consider the set

\begin{align} B = \{ iE_{j,k}| j,k = 1,2,...,n\} \end{align} where $E_{j,k}$ is the elementary matrix whose $(j,k)$th entry is $1$ and all others entry are zero.

Now It is straight forward to verify that $B$ is linearly independent over $\mathbb{R}$ and form a basis for the given subspace. Clearly dimension of the subspace is $n^2$.