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Rank-one updates intuition
A rank-one matrix is an outer product mathbfumathbfv^T: one column direction times one row...
Shape tells codomain and domain sizes first
An mtimes n matrix defines a linear map A:mathbbR^ntomathbbR^m. Rows count outputs; columns count...
Tall matrices: stacking measurements
Each row of a tall matrix can encode one sensor reading or one linear constraint. When m>n,...
Wide matrices: fat unknowns, short equations
When m<n, you have more unknowns than scalar equations. Either many free variables remain or...
Same intrinsic rank, different packaging
Rank is intrinsic: you can embed mathbbR^2 into mathbbR^100 with a tall skinny matrix while keeping...
Preview: duality links rows to linear functionals
Rows act on column inputs even before the dedicated dot-product chapter: each row defines a linear...
Handoff to dot products and taller geometry
Later chapters revisit inner products to reinterpret rows geometrically and to build orthogonal...
Dot product: length-times-shadow story
The standard dot product mathbfucdotmathbfv measures how far mathbfv extends along mathbfu,...
Duality: row vectors eat column vectors
Matrix-vector multiplication Amathbfx has a column picture and a row picture. The row picture says:...
Projections and least-squares bridge
Projecting mathbfb onto a subspace W picks mathbfpin W that minimizes |mathbfb-mathbfp|. The error...
Dual bases vs orthonormal trick
For a general (possibly non-orthonormal) basis mathbfb_1,ldots,mathbfb_n, the dual basis...
Rows as constraints in linear systems
Each row of Amathbfx=mathbfb is a scalar constraint: mathbfa_i^Tmathbfx=b_i. Geometrically, the set...