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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...
Vector addition: head-to-tail choreography
To add mathbfu and mathbfv, draw mathbfu, place the tail of mathbfv at the tip of mathbfu, and the...
Scalar multiplication: stretch, squeeze, and flip
Scaling a vector by c rescales its length by |c| and flips direction when c<0. The chapter...
Linear combinations: scale, then add, same recipe in every dimension
A linear combination of vectors mathbfv_1,ldots,mathbfv_k is any sum...
Linear independence: no hidden redundancy in the generator list
Vectors are linearly independent when the only combination giving mathbf0 is the trivial one with...
Coordinates depend on basis; geometry does not
Two observers can use different skewed grids while agreeing on the same arrows in space. Their...
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...
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...
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...
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...
Transformations in alternate coordinates sandwich
If M_textyours is the matrix of a linear map in your coordinates, the same geometric transformation...