# Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$

Could you help me with the following problem?

My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V \rightarrow \mathbb{R}$ such that $h(v)=\varphi(v,v)$.

Could you tell me how, based on that definition, I can prove that a quadratic form on $V=\mathbb{R}^n$ is a homogeneous polynomial of degree $=2$?

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Choose a basis for $V$, call it $\{v_1,\dots,v_n\}$. Then leting $v = \sum_{i=1}^n a_i v_i$ $$h(v) = \varphi(v,v) = \sum_{i=1}^n \sum_{j=1}^n a_i a_j \varphi(v_i,v_j).$$ Therefore $h(v)$ is a polynomial whose terms are $a_i a_j$ (i.e. degree $2$ in the coeficients of $v$) and the coefficient in front of $a_i a_j$ is $\varphi(v_i,v_j)$. This means $h(v)$ is an homogeneous polynomial of degree $2$ in the variables $\{a_1,\dots,a_n\}$.