[Math] Dimension of vector space of 2×2 skew symmetric matrices

Question

I had a question about the dimension of this subspace. This was related to a problem that had a case of n x n matrices, but I accidentally read it as the special case of 2×2, but never the less the answer to this question should help me with the general case.

Question

Consider the set of all skew-symmetric matrices of M_{2×2}, call it W. A matrix A is an element of this set provided that -A = A^{t}. Find a basis for W, what is the dimension of W?

Attempted Answer

If we consider A = \begin{pmatrix}
a &b \\
c&d
\end{pmatrix}

Then A^{t} = \begin{pmatrix} a & c \\ b & d\end{pmatrix}

So for a matrix to be in our subspace, we would have

\begin{pmatrix}-a & -b \\ -c & -d\end{pmatrix} = \begin{pmatrix}a & c \\ b & d\end{pmatrix}

Which implies that -a = a and -d = d, so therefore a and d are 0 (won't bother to show) and that b = -c.

Therefore the resulting set of matrices will look like \begin{pmatrix}0 & -c\\c & 0\end{pmatrix}

This is where I run into trouble. I'm not sure if the dimension of this subspace is 1 or 2. If I factor the c out, then I have the matrix \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}

and by taking any scalar multiple of this matrix, I can create any matrix in the subspace of 2×2 skew symmetric matrices. It's also linearly independent so it should be a basis.

My confusion arrises in that I could also decompose the matrix further into the two matrices

\begin{pmatrix}0 & -c \\ 0 & 0\end{pmatrix}

and

\begin{pmatrix} 0 & 0 \\c & 0\end{pmatrix}

These are both linearly independent vectors and span the subspace, so it's a basis, and I was wondering if that means the dimension of the subspace should be 2. But I wonder if the fact that the choice of c affects both matrices means that this decomposition isn't necessary and that since you can't choose two different weights for these matrices that the dimension of the subspace must in fact be one.

Sorry that was a long post, and maybe not the cleanest, still getting used to Latex. Thanks for any help.

Answer 1

$\begin{bmatrix}0 & -c \\ 0 & 0\end{bmatrix}$ is not a...