3.3.5

Let V be the space of symmetric nxn matrices with entries from a field $$\mathbb{F}$$. Construct a set of $$(n^2 + n)/2$$ symmetric matrices as follows: for $$i$$ satisfying $$1 \le i \le n$$, let $$S^{i,i}$$ be the matrix with a 1 on the i'th diagonal entry (that is, the (i,i) entry), and 0 elsewhere. There are n such matrices. Now for i,j satisfying $$1 \le i < j \le n $$, let $$S^{i,j}$$ be the matrix with a 1 on the (i,j) and (j,i) entry, and 0 everywhere else. There are $$n(n-1)/2$$ such matrices, so altogether, $$\{S^{i,j} : 1 \le i \le j \le n \}$$ forms a set of $$n(n+1)/2$$ symmetric matrices. We claim this set forms a basis for V.

The matrices in this set span V, for given any symmetric matrix A in V, denote the (i,j) entry of A by $$A_{ij}$$, then we can decompose A into a linear combination of the $$S^{i,j}$$ matrices by writing $$A = A_{11}S^{1,1} + A_{12}S^{1,2} + A_{22}S^{2,2} + ... + A_{nn}S^{n,n}$$, so that A is in the span of the matrices in this set. They are also linearly independent, for letting the $$ A_{ij}$$ be chosen such that $$A_{11}S^{1,1} + A_{12}S^{1,2} + A_{22}S^{2,2} + ... + A_{nn}S^{n,n} = 0_M$$, where $$ 0_M$$ denotes the zero matrix, the decomposition shows that all the coefficients $$A_{ij}$$ must be 0, thus we have a linearly independent spanning set of matrices, forming a basis for V.