Given a set of vectors, we can compute the number of independent vectors by calculating the rank of the set, but my question is how to find a maximal linearly independent subset. Thanks!
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One method is:
Another method is:
When step 3 instructs you to stop, $B$ contains a maximal linearly independent subset of $A$.
Form a matrix whose columns are the given vectors. Do row reduction to bring it to reduced form. In each non-zero row of the reduced form, circle the leftmost non-zero entry. The columns in the original matrix that correspond to columns in the reduced matrix with a circled entry - they form a maximal linearly independent set.