I got the following problem:
Let $S=\{A_1,A_2,...,A_k\} \subseteq \mathbb{M^R}_{n\times n}$ be a linear independent set of $k$ real $n \times n$ matrices with respect to the standard matrix inner product $\lt A,B \gt = trace(B^T A)$
Show that the matrix $U \in \mathbb{M^R}_{k \times k}$ defined by
$U = [trace({A^T}_i A_j)]_{1 \le i,j \le k} $ invertible
I tried to show that $U$ invertible but I failed so far. Thanks...
Check my comment and suppose we have a set $\;\{u_1,...,u_n\}\subset V\;,\;\;\dim V=n\;$ , and we have an inner product $\;\langle,\rangle\;$ defined on it.
Define $\;a_{ij}:=\langle u_i,u_j\rangle\;$ and let
$$A:=\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\ldots&a_{2n}\\\ldots&\ldots&\ldots&\ldots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix}$$
Suppose $\;A\;$ is singular $\;\iff\;$ there exists a column linearly dependent on the preceeding ones, say
$$\begin{pmatrix}a_{1k}\\a_{2k}\\\ldots\\a_{nk}\end{pmatrix}=\sum_{i=1}^{k-1}c_i\begin{pmatrix}a_{1i}\\a_{2i}\\\ldots\\a_{ni}\end{pmatrix}\iff$$
$$\forall\,1\le m\le n\;\;,\;\;a_{mk}=\sum_{i=1}^{k-1}c_ia_{mi}\iff\langle u_m,u_k\rangle=\sum_{i=1}^{k-1}c_i\langle u_m,u_i\rangle\iff$$
$$\iff \langle u_m\,,\,u_k-\sum_{i=1}^{k-1}c_iu_i\rangle=0$$
Well, try now to take it from here...