Everything in the MA103 mini block on one page — and thirty drills to make it automatic
WPR 1 provided no formula sheet, so assume WPR 2 will not either. This is the list to memorize. Print it, cover the right-hand side, and see how much you can reproduce from the name alone.
The drills at the bottom are deliberately mechanical. Speed on these is what buys you time for the interpretation questions, which is where the points actually go.
Lessons 23–26 · Reading M.1
Always ≥ 0. Squaring destroys sign, so negative components help exactly as much as positive ones.
The result is a scalar. Both vectors must have the same number of components.
Range −1 to 1. Measures direction only — immune to scaling either vector by a positive number.
Positive dot product → acute. Negative → obtuse. Zero → exactly 90°.
Perpendicular, 90°, cos θ = 0. One computation settles it.
One is a scalar multiple of the other. Equivalently cos θ = ±1. Negative k means antiparallel.
A weighted blend, computed component by component. It lands between the two vectors, nearer the heavier weight.
Put every feature on a common 0–1 scale before any distance or similarity work, or the largest-range variable dominates.
Lessons 27–32 · Reading M.2
Rows first, always. A column vector is n×1.
Swaps rows and columns. On a data matrix it switches "per observation" to "per variable."
Requires identical dimensions. Scalar multiplication touches every entry.
Inner dimensions must match; the result takes the outer ones. Entry (i,j) = row i of the left · column j of the right.
In general. Often one order is not even defined. Never reorder a product.
For [ a b ; c d ]. Nonzero ⇔ invertible.
Swap the diagonal, negate the off-diagonal, divide by the determinant. Check with AA−1 = I.
Only when det ≠ 0. Coefficients become A, constants become b⃗.
Ri ↔ Rj; Ri ← kRi with k ≠ 0; Ri ← Ri + kRj. These three, nothing else.
Pivot in every coefficient column → unique. Row [0 … 0 | nonzero] → none. Fully zero row → infinitely many.
Rows are scalar multiples. Geometrically: parallel lines — either never meeting, or the same line. Row reduce to find out which.
Simultaneous dot products — one weighted score per observation, all at once. Scaling w⃗ scales every score and never changes the ranking.
Cover the answers. Aim for under five seconds each. Use the Drill answers switch in the bar above to check.
Given A 2×2, B 2×3, C 3×2, D 3×3, u⃗ 2×1, v⃗ 3×1. State defined or undefined, and the dimensions.