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 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 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.
Read note →Beginning Chapter 3 with an intuitive understanding of vector spaces and subspaces: real components, R^2, closure under linear combinations, why y=2x forms a subspace but y=2x+1 does not, and how constraints reduce independent freedom.
Read note →A practical path through transpose and permutation matrices: why transpose is needed, how it connects rows and columns, why (AB)^T reverses order, why permutation matrices satisfy P^{-1}=P^T, and how left multiplication PA actually rearranges rows.
Read note →Gaussian elimination is easy to demonstrate on small matrices, but real systems may contain thousands or millions of unknowns. This note follows the question of how computers solve large Ax = b systems, why LU makes repeated solutions efficient, and why zero or tiny pivots lead naturally to row exchanges and PA = LU.
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