Basis and Dimension of 3x3 Antisymmetric Matrices

The problem asks us to find a basis and the dimension of the space of antisymmetric matrices. A matrix is antisymmetric if its transpose is equal to .

Let be a matrix:

Its transpose is:

The condition for to be antisymmetric is .

Comparing the elements of the matrices, we get the following equations:

This means that all diagonal elements of an antisymmetric matrix must be zero.

For the off-diagonal elements:
4.
5.
6.

And also:
7. (same as 4)
8. (same as 5)
9. (same as 6)

So, an antisymmetric matrix has the form:

We can express this matrix as a linear combination of simpler matrices. Let , , and .

The matrices

form a basis for the space of antisymmetric matrices. These matrices are linearly independent because if , then

which implies , , and .

Since there are 3 matrices in the basis, the dimension of the space of antisymmetric matrices is 3.

In general, for an antisymmetric matrix, the diagonal elements are 0. For the off-diagonal elements, . This means that the elements above the main diagonal determine the elements below the main diagonal. There are elements above the main diagonal. Thus, the dimension of the space of antisymmetric matrices is .
For , the dimension is .

The final answer is .