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Linear combinations: scale, then add, same recipe in every dimension
A linear combination of vectors mathbfv_1,ldots,mathbfv_k is any sum...
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...
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...
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...
Guest bridge: from classical optimization to generative stacks
Welch Labs extends the playlist's visual vocabulary toward image and video synthesis. The same...
Matrix-vector multiplication: combine columns with input coefficients
The product Amathbfx linearly mixes columns of A using entries of mathbfx as weights. Outputs stay...
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)....
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...
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...