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Non-commutativity: order usually matters
Even when both AB and BA exist, they often differ. Different pipelines deform space differently...
Column picture of $AB$: $A$ times each column of $B$
Column j of AB equals A times column j of B. Thus the column space of AB lies inside the column...
Chapter close: language hygiene before transformations
The narrative tightens vocabulary: distinguish vectors , points , coordinates , and bases before...
Geometric story before cofactor expansions
Pictures come before Laplace expansions here [@1]. Understand collapse events first: projections...
Same intrinsic rank, different packaging
Rank is intrinsic: you can embed mathbbR^2 into mathbbR^100 with a tall skinny matrix while keeping...
Transformations in alternate coordinates sandwich
If M_textyours is the matrix of a linear map in your coordinates, the same geometric transformation...
Preview: duality links rows to linear functionals
Rows act on column inputs even before the dedicated dot-product chapter: each row defines a linear...
Parsing the "backwards" feeling
Students often feel the change-of-basis story runs backward. The resolution is to separate geometry...
Trace and determinant stay similar
Similarity B^-1MB preserves trace and determinant even when individual entries scramble [@2]. That...
Worked mental checklist for change of basis
A reliable workflow: identify columns of B, confirm invertibility, then multiply or sandwich as the...
Eigenbasis and diagonalization
When eigenvectors form a basis, A = B D B^-1 with diagonal D listing eigenvalues on the diagonal...
Handoff to dot products and taller geometry
Later chapters revisit inner products to reinterpret rows geometrically and to build orthogonal...