In the realm of Lie algebras, matrices can serve as a powerful tool to represent elements of the algebraic structure. Let’s take a closer look at matrix representation using the matrix \(E_{f}^{i}\). This matrix is special in that all of its elements are zero except for one: the \(i, j\) component, which is set to 1.
A system of such matrices \(E_{f}^{i}\) forms a basis for the algebra \(G\mathcal{C}(n, \mathbb{R})\). Since \(i\) and \(j\) can each vary from 1 to \(n\), there are a total of \(n^2\) possible matrices, aligning perfectly with the dimension of the algebra \(G\mathcal{C}(n, \mathbb{R})\). This shared dimension confirms that these matrices can form a complete basis.
Additionally:
- They are linearly independent. Any scalar multiplication or addition of different \(E_{f}^{i}\) matrices produces another matrix within the set with only one non-zero entry.
- Each matrix contributes to spanning the vector space since it is constructed from unique position pairings \((i, j)\).
Thus, these matrices effectively capture the structure of the given algebra through their straightforward but powerful representation.