How to construct a basis using a given dependent set of vectors using Gaussian elimination?

Question

Suppose we have a given set of vectors $\{a_1,a_2,...,a_k\}$ in $\mathbb R^n$ where $k>n$.How do I find a basis out of the given vectors i.e. to find a subcollection of this set that forms a basis.I want to do this using Gaussian elimination.But row reduction may change all the vectors and give me a basis that is not a subcollection of the given one.How to avoid that?I have heard about considering something like 'pivot' position but I do not understand how they do it.Can someone tell me about making a basis using pivot positions and what pivot is?Please illustrate through examples.

Answer 1

You can use gaussian elimination to check what vectors are linear independent but you need to keep the original vectors that are linear independent to use like basis.

for example if you have $\operatorname{Gen}(S) = \operatorname{span}\{(1,1,0),(1-1,1),(2,0,1)\}$

$$\left( {\begin{array}{*{20}{c}}1&1&2\\1&{ – 1}&0\\0&1&1\end{array}} \right)\;\mathop \to \limits_{{F_2} \to {F_1} – {F_2}} \;\;\;\left( {\begin{array}{*{20}{c}}1&1&2\\0&2&2\\0&1&1\end{array}} \right)\;\;\;\mathop \to \limits_{{F_3} \to {F_2} – 2{F_3}} \;\;\;\left( {\begin{array}{*{20}{c}}1&1&2\\0&2&2\\0&0&0\end{array}} \right)$$

The basis for subspace $S$ could be ${(1,1,0),(1-1,1)}$ or ${(1,1,0),(2,0,1)}$

You can keep any linearly independent combination of vectors from the set.