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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...
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...
Coordinates as translators between vectors and tuples
Once a basis is fixed, every vector gains a numeric fingerprint: the ordered coefficients. Changing...
Chapter synthesis: independence controls redundancy, span controls reach
The chapter pairs two audits: redundancy (dependence) and coverage (span). Basis vectors pass both...
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)....
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...
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...
Non-invertible maps collapse at least one direction
Dependent columns mean several inputs share outputs; the kernel captures those differences....
Packing the chapter for matrix multiplication next time
The narrative ends poised for composition: you now read Amathbfx as geometry encoded in columns....
Composition: the matrix nearest the vector acts first
If S applies first then T, the composite on column vectors is TS, so (TS)mathbfx = T(Smathbfx)...