Journal

Tag: linear-independence

Long-form notes, Qiskit experiments, and applied optimization thinking.

September 25, 2026•Learning Notes•5 min read

Why Linear Dependence Is Tested With Zero: Redundancy, Span, and the Nullspace

Why linear dependence is tested by a nontrivial combination equaling zero, why arbitrary coefficients do not produce every vector, how span limits what can be produced, and how dependence connects directly to the nullspace.

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September 10, 2026•Learning Notes•6 min read

Making Nullspace and Linear Independence Intuitive: Redundancy and Invisible Input Directions

A question-driven explanation of nullspace and linear independence, moving beyond Ax = 0 to the mental pictures of redundant vectors and input changes that a matrix cannot detect.

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