Vectors in R^n

1A vector is an ordered point or arrow

A vector in \mathbb{R}^n is an ordered collection of n real numbers. The order is not decoration: the first slot and the second slot play different roles, so the pair (2, -1) and the pair (-1, 2) are different vectors.

In geometry the same ordered pair can name a landing from the origin — two steps right and one step down — or name the gold arrow that makes that move. The sentence around the pair chooses the role; the numbers themselves do not.

To go from one point to another, subtract. With start P=(2,-1) and end Q=(8,3), the displacement is Q-P=(6,4). Distance is the length of that arrow; the midpoint averages each slot and is a point, not an arrow.

Read a coordinate vector

1. Fix axesCoordinate 1 is how far right from the origin; coordinate 2 is how far up, or down if negative. Keep that order for the whole lesson.
2. Point viewRead (2, -1) as the landing: two steps along coordinate 1, then one step down along coordinate 2.
3. Arrow viewThe same pair names the move from the origin to that landing. From P to Q, the displacement is Q-P, end minus start.

2Add and scale component by component

Vector addition combines matching coordinates, while scalar multiplication applies the same multiplier to every coordinate. These two operations are the grammar of linear algebra — slot-by-slot bookkeeping on a list, not a new kind of arithmetic.

Picture a warehouse robot stuck mid-path at floor position x=(4, 1, 6) metres: aisle, bay, and shelf height in that fixed order. A planner chose nudge d=(2, -3, 1), but the operator wants only half of that path for a careful approach. Scale d first, then add — and keep the same slot order, or the arithmetic silently mixes aisle metres with bay metres. The same algebra later updates model features; the story here is the floor move.

Move the robot carefully

1. Match slotsLine up aisle, bay, and height so each nudge component sits under the coordinate it is meant to change.
2. Scale firstMultiply every component of d by the scalar — here 0.5 for the careful half-step.
3. Add secondAttach the scaled nudge tip-to-tail and add component by component to get the new floor position.

3A dot product is a weighted sum

The dot product takes two equal-length vectors and multiplies matching coordinates before adding the products. It is not a new kind of multiplication table; it is a compact way to say weighted sum — compress two lists into one score.

Picture a plant control room waiting on one alarm number. Sensors already report readings x=(1, 4, 0.5): pressure, vibration events, and heat utilisation in that fixed order. The operator chose weights w=(2, -1, 3) — boost pressure, penalise vibration, boost heat. Pair matching slots, multiply, then add. Keep the same order, or the arithmetic silently mixes a vibration reading under a heat weight. The same multiply-add later scores an applicant or one neuron; the story here is the alarm score.

Compute the alarm score w\cdot x

1. PairLine up each weight w_i only over the sensor reading x_i in the same slot — pressure with pressure, vibration with vibration, heat with heat.
2. MultiplyCompute each matched product, preserving negative signs (the vibration penalty stays minus).
3. AddSum the products to get one scalar alarm score — not another vector.

Coding lab. Compute a vector score with NumPy runs in the app, with checks on your output.

4Projection finds the course shadow; cosine reads the angle

A nav mission desk needs the course shadow of planned heading u along reference heading v. Projection is that shadow: \mathrm{proj}_v(u)=((u\cdot v)/(\|v\|^2))v. Speak the pieces: the numerator is the dot product, the denominator is length squared — keep it squared — and the last step multiplies that one scalar by every slot of v. The foot F from the origin is exactly that projection vector; the residual u_\perp=u-\mathrm{proj}_v(u) is the leftover from F to the tip of u. Derive the foot from the scalar times v; never guess it from a sketch. The same formula works in \mathbb{R}^n when no honest picture of the drop exists.

Once that adjacent shadow is known, cosine keeps only the angle. Adjacent is \|\mathrm{proj}\|, hypotenuse is \|u\|, so \cos\theta=\|\mathrm{proj}\|/\|u\|. The algebraic twin (u\cdot v)/(\|u\|\|v\|) returns the same number. For u=(3,4) and v=(4,0), the desk gets \cos\theta=0.6 and \theta\approx53.1^\circ — partial alignment: closer than perpendicular, not the same heading. Raw needle length is not alignment; cosine divides size out on purpose.

Project by algebra, then read cosine

1. Dot and scaleCompute u\cdot v and \|v\|^2, form the scalar (u\cdot v)/(\|v\|^2), then multiply by v to get \mathrm{proj}_v(u)=F.
2. ResidualSubtract u_\perp=u-\mathrm{proj}_v(u). In 2D you may draw afterward; in higher dimensions keep the same formula without a picture.
3. CosineRead \cos\theta=\|\mathrm{proj}\|/\|u\| or the twin (u\cdot v)/(\|u\|\|v\|) — same alignment number.

5Basis vectors give coordinates their meaning

A basis is the set of reference directions used to describe vectors. Coordinates are the amounts of those directions. In the standard basis of \mathbb{R}^3, those directions are e_1=(1,0,0), e_2=(0,1,0), and e_3=(0,0,1) — each e_i turns on exactly one coordinate and leaves the others at zero.

Picture a CNC machine bed stuck on a destination ticket p=(4,-2,7). CNC means a computer-guided cutting or milling machine — like a computer-guided mill or a 3D-printer bed. Three numbers sit on the job sheet, but they are useless until the bed's reference directions are named. Fix the standard axes first; then the ticket means four of e_1, minus two of e_2, and seven of e_3. Scale each basis vector, add the three pure-axis pieces, and the sum recovers p. Change the basis and the same written list lands somewhere else on the bed.

Rebuild the CNC destination from the standard basis

1. Name axesWrite the standard basis: e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1). Until those directions are fixed, the ticket is only a list.
2. Read amountsIn that basis, p=(4,-2,7) means four copies of e_1, negative two of e_2, and seven of e_3.
3. Scale then addScale each basis vector, then add the pure-axis pieces: 4e_1=(4,0,0), -2e_2=(0,-2,0), 7e_3=(0,0,7), and their sum is p.

6One data row is a vector

Open a study-log spreadsheet. One student row has columns intercept, hours studied, and past score filled as 1, 6, and 0.8. Those three cells in a fixed column order are already a feature list — and that list is a vector in \mathbb{R}^3. With n feature columns, the same reading is a vector in \mathbb{R}^n — \mathbb{R}^n means the space of fixed-order lists of n real numbers.

Column order is coordinate order: slot one becomes x_1, slot two becomes x_2, and so on. Pack the readings as x=(1,6,0.8). Once a row is a vector, earlier operations become machine-learning (ML) operations: add an update, score with a dot product, or compare rows by distance or direction. Swap the column order and the same digits become a different vector, so every later weight that expected intercept first silently mis-fires.

Build the feature vector from the study-log row

1. Fix columnsRead headers in order: intercept, hours studied, past score. Column order is coordinate order — whatever sits in slot one becomes x_1.
2. Read slotsFrom the same row: x_1=1 (intercept), x_2=6 (hours studied), x_3=0.8 (past score).
3. Pack and nameWrite x=(x_1,x_2,x_3), here (1,6,0.8) — a vector in \mathbb{R}^3. With n columns the same idea is a vector in \mathbb{R}^n. Keep the order fixed or later weights mis-fire.