Let $\mathbb{K}$ be an arbitrary field with $char(\mathbb{K})\neq 2$ and let $a_1, \ldots ,a_n$ be Elements of $\mathbb{K}$.
I want to proof the following equivalence of $n-$quadratic forms in diagonal form over $\mathbb{K}$: $ \begin{align} [a_1, \ldots ,a_n] \simeq [c_1^2a_1, \ldots , c_n^2a_n], \end{align} $ where $c_1, \ldots , c_n \in \mathbb{K}^* \forall n \in \mathbb{N}$.
According to Lam's Book $\textit{Introduction to quadratic forms}$ (p. 1-2) it suffices to find an "$\textit{invertible matrix}$ $C \in GL_n(\mathbb{K})$ $\textit{such that}$ $ g(C\cdot X)=f(X)$. (1) $\textit{This means there exists a nonsingular, homogenuous linear substitution of the variables}$ $X_1, \ldots ,X_n $ $ \textit{that takes the form}$ $g$ $\textit{to the form}$ $f$."
Now I was going to define the isometry
$C:=diag(\sqrt{c_1},\ldots ,\sqrt{c_n})$.
Clearly $C \in GL_n(\mathbb{K})$ and (1) would be established. But I cannot really say if this substitution is "homogenuous" as understood in Lem's sense.
Can someone give me insight into this?