A quadratic form in an $m$ -dimensional  real vectorspace $V$, is a bilineal map $V\times V\to\mathbb{R}$ which can be determined by a $m\times m$-matrix $A$ via

$(v,w)\mapsto v^TAw$

If we are allowed to write $g(v,w)=v^TAw$ we can find that he (or she) possibly satisfy

1. $g(v,w)=g(w,v)$,  property dubbed symmetry
2. $g(v,v)\ge0$, positive-definiteness
3. $g(v,v)=0$ iff $v=0$, non-degeneracy

In case of affirmatively both three are satisfy $g$ is called a metric on $V$ and the pair

$(V,g)$

is called (real) Euclidean vectorspace…

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