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Scalar multiplication: stretch, squeeze, and flip
Scaling a vector by c rescales its length by |c| and flips direction when c<0. The chapter...
Cross-training for cross products next chapter
Dot products measure alignment along a direction; cross products (next chapter) build perpendicular...
Coordinates as linear combinations of basis arrows
Writing (x,y) as xhatmathbfi+yhatmathbfj reveals the secret grammar: coordinates list the scalars...
Tall matrices: stacking measurements
Each row of a tall matrix can encode one sensor reading or one linear constraint. When m>n,...
Linear combinations: scale, then add, same recipe in every dimension
A linear combination of vectors mathbfv_1,ldots,mathbfv_k is any sum...
Orientation: sign of determinant
The sign of det A tells you whether orientation is preserved or reversed [@2]. A two-dimensional...
Computing inverses conceptually with row operations
Augment [Amid I] and row-reduce. When A is invertible, the reduced form becomes [Imid A^-1] [@2]....
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...
Geometric definition in $\mathbb{R}^3$
The cross product mathbfatimesmathbfb in mathbbR^3 is a vector perpendicular to both mathbfa and...
Bilinearity and algebraic determinant template
Cross product is bilinear: linear in each argument when the other is fixed. Distribution rules...
Cross products set up transformed versions next chapter
The next chapter asks how T(mathbfatimesmathbfb) relates to T(mathbfa)times T(mathbfb) for a linear...