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

Question

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

Answer 1

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 $\mathbb{R}^n/W$.