# Algorithm to find a basis of a quotient space $R^n/R^m$.

I have a set of $$m$$ vectors $$\{x_i\}$$, $$x_i \in R^n$$. How can I obtain a basis for $$R^n/span(\{x_i\})$$?

Find a basis of $$\text{span(\{x_i\})}=:W$$, say $$\{x_1,x_2,... ,x_k\}$$, and a basis of $$\mathbb{R}^n$$ of the form $$\{x_1,x_2,...,x_k\}\cup\{y_1,y_2,...,y_{n-k}\}$$. Then the classes $$y_j+W, 1\leq j\leq n-k$$, are a basis of $$\mathbb{R}^n/W$$.
• Yes, but how do we obtain the $\{y_i\}$?. In 3D (n=3), if m=2, we can take the cross product. If m=1, and $x_0=[a,b,c]$ then take $y_0=[−b,a,0]$ and $y_1=x_0 \times y_0$ (with some linear independence assumptions). Does this algorithm generalize? – Scott Oct 19 '18 at 8:56