When are vectors linearly independent?

A (finite) set of vectors $ v_1, v_2...v_m $is said to be linearly independent if and only if the equality $ k_1v_1+k_2v_2+...k_mv_m=0 $ is true exactly when all the k values are 0.

This is equivalent to saying you can't come up with any linear combination of $ v_1 $ and $ v_2 $ that equals $ v_3 $, or $ v_1...v_3 $ that equals $ v_4 $... or $ v_1...v_{m-1} $ that equals $ v_m $.

If a set of vectors are not linearly independent, then they are linearly dependent.

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett