Journal
Long-form notes, Qiskit experiments, and applied optimization thinking.
A step-by-step study of Strang's parameter-dependent rank example, showing how pivots identify pivot variables, why nonpivot variables are free, and why c=4 causes the rank to drop and the nullspace to gain an extra direction.
Read note →An intuitive study of matrix rank, why pivots measure independent information, how special nullspace solutions expose redundant columns, and why rank-one matrices have the form A=uv^T.
Read note →Why perfect entanglement correlations naturally suggest predetermined answers, what locality and realism mean, and why Einstein was led to the idea of hidden variables.
Read note →A step-by-step study of Bernhardt's example showing how an initially unentangled pair of qubits becomes the Bell state through a CNOT gate, beginning with the tensor product and following the calculation explicitly.
Read note →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.
Read note →How row reduction exposes free variables, how free variables produce special solutions, and why the entire nullspace is the span of those special solutions.
Read note →A step-by-step study of Strang Chapter 3 Section 3.2 showing how elimination reveals redundant equations, free variables, special solutions, and finally the complete nullspace.
Read note →A learning checkpoint in the study of the Schrödinger equation: what the Hamiltonian represents, why energy is represented by an operator, and why the discussion should pause until eigenvalues and eigenvectors have been studied in linear algebra.
Read note →A question-driven introduction to quantum evolution, wavefunction collapse, linearity, and why understanding the Schrödinger equation is necessary before claiming that ordinary quantum evolution preserves superposition.
Read note →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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