J

Equation

Solution Space

0

${\displaystyle {\sum }_{q=0}^0}{\beta }_q{\gamma }_{\left(k,q\right)}={\beta }_0{\gamma }_{\left(k,0\right)}=0$

$\Rightarrow {\beta }_0=0$ i.e. {0} the point

1

${\displaystyle {\sum }_{q=0}^1}{\beta }_q{\gamma }_{\left(k,q\right)}={\beta }_0{\gamma }_{\left(k,0\right)}+{\beta }_1{\gamma }_{\left(k,1\right)}=0$

$\Rightarrow \varphi \left({\beta }_0\mathrm{,}{\beta }_1\right)=0$ i.e. a subspace of ${ℝ}^2$

2

${\displaystyle {\sum }_{q=0}^2}{\beta }_q{\gamma }_{\left(k,q\right)}={\beta }_0{\gamma }_{\left(k,0\right)}+{\beta }_1{\gamma }_{\left(k,1\right)}+{\beta }_2{\gamma }_{\left(k,1\right)}=0$

$\Rightarrow \varphi \left({\beta }_0\mathrm{,}{\beta }_1\mathrm{,}{\beta }_2\right)=0$ i.e. a subspace of ${ℝ}^3$

$⋮$

$⋮$

$⋮$

n

${\displaystyle {\sum }_{q=0}^n}{\beta }_q{\gamma }_{\left(k,q\right)}=0$

$\Rightarrow \varphi \left({\beta }_0\mathrm{,}{\beta }_1\mathrm{,}\cdots \mathrm{,}{\beta }_n\right)=0$ i.e. a subspace of ${ℝ}^{n+1}$