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Duality: row vectors eat column vectors
Matrix-vector multiplication Amathbfx has a column picture and a row picture. The row picture says:...
Orthogonality identities
By construction, mathbfacdot(mathbfatimesmathbfb)=0 and mathbfacdot(mathbfbtimesmathbfa)=0: the...
Span: the flat region of everything reachable from a generating set
mathrmspanmathbfv_1,ldots,mathbfv_k is the set of all destinations produced by some choice of...
Chapter synthesis: independence controls redundancy, span controls reach
The chapter pairs two audits: redundancy (dependence) and coverage (span). Basis vectors pass both...
Linear independence: no hidden redundancy in the generator list
Vectors are linearly independent when the only combination giving mathbf0 is the trivial one with...
Basis: minimal spanning set, no junk directions, no missing directions
A basis for a subspace is an independent spanning family; order matters because coordinates list...
Matrices in the standard basis: columns are destination arrows
For T:mathbbR^ntomathbbR^m under standard bases, column j of A is T(mathbfe_j). That is the entire...
Linearity: additivity and scaling, no translation of space
A map T is linear if T(mathbfu+mathbfv)=T(mathbfu)+T(mathbfv) and T(cmathbfu)=cT(mathbfu)....
Matrix-vector multiplication: combine columns with input coefficients
The product Amathbfx linearly mixes columns of A using entries of mathbfx as weights. Outputs stay...
Examples: rotation, shear, projection, same rules, different columns
Animations rotate the basis, shear along an axis, or squash onto a line. Each example reads off as...
2×2 picture: area ratios
In mathbbR^2, columns mathbfa_1,mathbfa_2 span a parallelogram. Replacing mathbfa_1 by mathbfb...
Transition to change of basis formulas
Coordinate remapping reinterprets the same geometry in new language. Chapter 13 develops change of...