Everything in the MA103 mini block on one page, and fifty-nine drills to make it automatic
Gate 3 has come back and it was 100% computation: seven questions, 25 points, not one asking for a sentence. The Gate 3 mirror rebuilds it question for question. That confirms what the course director said in advance: WPR 2 will be "much more computational than WPR 1," and that cadets will have to "demonstrate their computational ability with vector and matrix operations" on top of interpreting their results. That makes this page the highest-leverage thing you can work.
WPR 1 provided no formula sheet, so assume WPR 2 will not either. The cards below are the list to memorize. Print the page, cover the right-hand side, and see how much you can reproduce from the name alone.
The 59 drills below are deliberately mechanical. Speed there is what buys you time for the interpretation questions, and this time the arithmetic is worth points in its own right.
Lessons 23–26 · Reading M.1
Always ≥ 0. Squaring destroys sign, so negative components help exactly as much as positive ones.
Divide every component by the magnitude. Keep it in fractions. Check: |v̂| = 1. Same direction, length thrown away.
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°. cos θ is not θ. Gate 3 scored stopping at the cosine as a major error.
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 b.
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 w scales every score and never changes the ranking.
Groups A–F are arithmetic; G–J are recognition. Do the arithmetic ones with pencil and paper, not a calculator. Use the Drill answers switch in the bar above to check.
Every one of these comes out a whole number. If you get a decimal, check your squaring.
Exact fractions, not decimals. Each answer should have magnitude 1.
Given A 2×2, B 2×3, C 3×2, D 3×3, u 2×1, v 3×1. State defined or undefined, and the dimensions.