1. ## Homogeneous equations

Let W be sub-space in dimension m of F^n (m < n). Show that there exist m - n homogeneous equations in n variables, so that-

W={(x_1,...x_n) | a_11*x_1+...+a_1n*x_n=0,...,
a_n-m*x_1+...+a_n-mn*x_n=0}

(x_1,...x_n) is column vector.

2. Originally Posted by Also sprach Zarathustra
Let W be sub-space in dimension m of F^n (m < n). Show that there exist m - n homogeneous equations in n variables, so that-

W={(x_1,...x_n) | a_11*x_1+...+a_1n*x_n=0,...,
a_n-m*x_1+...+a_n-mn*x_n=0}

(x_1,...x_n) is column vector.
Pick a basis for $\displaystyle W$ say $\displaystyle \{w_1,...,w_m\}$ and extend to a basis for $\displaystyle \mathbb{F} ^n$ say we add $\displaystyle \{ v_{1},...,v_{n-m} \}$ and take $\displaystyle T: \mathbb{F} ^n \rightarrow \mathbb{F} ^{n-m}$ (giving the later a basis $\displaystyle \{ u_1,...,u_{n-m} \}$) given by $\displaystyle T(w_i)=0$ and $\displaystyle T(v_i)=u_i$. We then have $\displaystyle \ker (T)=W$ and so picking a suitable matrix to represent this we get our equations.