Linear Independence Definition

Linear Algebra (Matrices) · Axiom Academy

When does a nontrivial mix of vectors collapse back to ? That single question separates redundancy from genuine new directions. 1. Dependence: a Nonzero Mix Returns to the Origin Vectors are linearly dependent when there is some choice of weights — not all zero — for which the combination lands exactly on : a NONTRIVIAL solution exists ⟹ dependent Watch three vectors that look unrelated. Walk along them head-to-tail with weights c_1=c_2=c_3=1 — and the chain closes back onto the origin . A nonzero mix reached , so the set is dependent. 2. Independence: Only the All-Zero Mix Reaches Drop the redundant vector. With just and , sweep the weights c_1,c_2 through every nonzero value. The chain tip traces a path — but it never touches the origin unless both weights are zero. the ONLY solution is the trivial one ⟹ independent Only gives . Each vector adds a new direction; no redundancy. Some nonzero mix gives . One vector is a combination of the others — a redundant direction. Linear independence comes down to one question: can a nonzero mix of the vectors reach ? Scroll up to replay either animation.

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