# Definition # Vector Space Matrices of [[dimension]] $n\times n$ with entries from a field $\mathbb{F}$ form a [[vector space]] denoted $M_n(\mathbb{F})$. Here, the binary operation of the vector space is normal matrix addition, and scalar multiplication is the usual multiplication of scalars by matrices. Using the [[Index Notation|index notation]], $ \begin{align} (A + B)_{ij} &\equiv A_{ij} + B_{ij},\\ (c A)_{ij} &\equiv cA_{ij} \end{align} $ For $\mathbb{F}=\mathbb{R}$, $M_n(\mathbb{R})$ is a real vector field, and $M_n(\mathbb{C})$ can be either a real or complex vector field. Two other vector spaces are $S_n(\mathbb{R})$ and $A_n(\mathbb{R})$, the vector spaces of symmetric and [[skew-symmetric matrix|antisymmetric]] matrices. Note that we have: $ \dim(A_n(\mathbb{R})) + \dim(S_n(\mathbb{R})) = \dim(M_n(\mathbb{R})) $ ## Basis A [[basis]] for $M_n(\mathbb{F})$ is: $ \{E_{ij}\}_{i,j = 1, \ldots, n} = (\delta_{ii'}\delta_{jj'})_{1\leq i', j' \leq n} $ For example, with $n = 3$, we have $ E_{23} = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} $ These are known as the *elementary matrices* and they form as basis. These are not special however, we could take: $ S_{ij} \equiv E_{ij} + E_{ji}, i \leq j $ or $ A_{ij} \equiv E_{ij} - E_{ji}, i < j $ as a basis, where the first set is symmetric matrices and the second is [[skew-symmetric matrix|antisymmetric]] matrices. Note that there are $n^2$ such matrices so that the [[dimension]] of $M_n(\mathbb{F})$ taken over the field $\mathbb{F}$ is $n^2$.