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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.

Rodrigo de Azevedo
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Kishalay Sarkar
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  • Use this {http://www.gregthatcher.com/Mathematics/ColumnSpaceCalculator.aspx} calculator. Each and every step will be cleared. – Siddhant Sep 21 '19 at 03:54
  • Take the matrix whose columns are $a_1,\ldots,a_k$. Reduce this matrix to row echelon form. Then the columns that are leading in this row echelon form correspond to columns from the original ${ a_1,\ldots,a_k}$ that form a basis for $\mathrm{span}{a_1,\ldots,a_k}$. – Minus One-Twelfth Sep 21 '19 at 09:07
  • But this basis might not be a subcollection of the original given set. – Kishalay Sarkar Sep 21 '19 at 12:17
  • https://math.stackexchange.com/questions/164016/how-to-find-maximal-linearly-independent-subsets do you understand the answer given here? it is very extensive – B.Swan Sep 21 '19 at 14:11
  • I searched in stack exchange for this question but the title is such that I could not find the answer,so it is should not actually be called duplicate. – Kishalay Sarkar Sep 21 '19 at 14:29
  • I vote for removing duplicate. –  Sep 21 '19 at 14:34

1 Answers1

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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.

José Carlos Santos
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Richard Lethbir
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