Span — definition: Foundations — Maths deep dive

Before you can read the parent note, you must own every symbol it throws at you. Below, each symbol is built from zero — plain words first, then a picture, then why the topic can't live without it. Nothing is used before it is defined.

0. What is a vector? (the very first arrow)

Picture it. The number on top is horizontal ("how far across"), the number below is vertical ("how far up"). The arrow is drawn from the origin to that tip.

Why the topic needs it. The span is "every place you can reach with arrows." So we must first know what a single arrow is and how to name it. Everything downstream is built out of these.

1. The origin and the zero vector $0$

Picture it. A dot at the centre of the page. "Staying home" — you never left.

Why the topic needs it. The parent note says "a span always passes through the origin." That statement is about $0$ : because you're always allowed to not move, the zero vector is always reachable. You cannot understand that sentence without owning this symbol.

2. Scalars, the letter $c$ , and $R$

Picture it. A number line stretching left and right forever. Picking $c$ means dropping a pin anywhere on it.

Why the topic needs it. "Scale, flip, add" — the scale and flip are exactly "multiply by a scalar." Because $c$ ranges over all of $R$ (including negatives and zero), your arrow can shrink to nothing, flip backwards, or stretch to any length. The parent's mistake box "span requires only positive scalars" is wrong precisely because $R$ includes negatives.

3. Scalar multiplication $c v$ — stretching one arrow

Picture it (all cases — this is the important part).

$c > 1$ : the arrow grows longer, same direction.
$c = 1$ : unchanged.
$0 < c < 1$ : shrinks toward the origin, same direction.
$c = 0$ : collapses to $0$ (the origin).
$c < 0$ : flips to point the opposite way, then scaled by $∣ c ∣$ .

Why the topic needs it. As $c$ sweeps through all of $R$ , the tip of $c v$ traces out a whole line through the origin. That single fact is why "the span of one vector is a line." Watch the yellow tip in the figure slide along the line as $c$ changes.

4. Vector addition $u + v$ — tip-to-tail

Picture it. Draw $u$ . Then start $v$ at the tip of $u$ (slide it over without rotating). Where $v$ ends is the tip of $u + v$ . This is the tip-to-tail rule.

Why this tool and not another? We need a way to combine two arrows into one reachable spot. Coordinate-wise addition is the only rule that matches the physical "walk along $u$ , then walk along $v$ " motion — the diagonal of the parallelogram. Scaling changes length; addition combines directions. The span needs both, because with only one move you'd be stuck.

5. Linear combination — scale, flip, add, all at once

Picture it. Stretch each arrow by its own scalar, then chain them tip-to-tail. The final tip is one point in the span. Change the scalars → land somewhere new.

Why the topic needs it. This is the raw material of a span. The parent's definition $span {v_{1}, \dots, v_{k}} = {c_{1} v_{1} + \dots + c_{k} v_{k} ∣ c_{i} \in R}$ is literally "the set of all linear combinations." See Linear combination for its own deep dive.

6. Set-builder notation ${\cdot ∣ \cdot}$

Picture it. A cookie-cutter (the recipe $c v$ ) stamping out infinitely many cookies as you vary $c$ . The whole pile of cookies is the set.

Why the topic needs it. A span is an infinite collection. Braces are how we say "all of these at once" without listing them. The parent's mistake "the span is just ${v_{1}, \dots, v_{k}}$ " confuses the few generators inside the braces with the infinite recipe-output the braces describe.

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mean as an arrow?

Go 2 steps right, 1 step up; the arrow points from the origin to that tip.

What is the zero vector

0

and where does its tip sit?

The arrow that goes nowhere,

[00]

; its tip is on the origin.

What does

c \in R

allow

c

to be?

Any real number at all — positive, negative, fractional, or zero.

Geometrically, what does

c v

trace as

c

runs over all of

R

A full line through the origin in the direction of

v

(both ways).

How do you add two vectors, and what does it look like?

Add matching coordinates; picture laying

v

tip-to-tail after

u

, giving the parallelogram diagonal.

What is a linear combination in words?

Scale each vector by its own scalar, then add them all up.

What does the bar

∣

mean inside

{\cdot ∣ \cdot}

"Such that" — left side is the recipe, right side is the condition on the ingredients.

What are the three rules a subspace must satisfy?

Contains the origin, closed under addition, closed under scaling.

What does

det \neq = 0

tell you about two vectors in the plane?

Their parallelogram has real area, so they span all of

R^{2}

Connections

Linear combination — the recipe built here becomes the building block of every span.
Subspace — the shape a span always turns out to be.
Determinant — the number that detects "fills the whole plane."
Linear independence — decides whether extra arrows add new directions (next deep dive).
Basis · Dimension · Column space — where these foundations lead once span is mastered.

Foundations — Span — definition

0. What is a vector? (the very first arrow)

1. The origin and the zero vector $0$

2. Scalars, the letter $c$ , and $R$

3. Scalar multiplication $c v$ — stretching one arrow

4. Vector addition $u + v$ — tip-to-tail

5. Linear combination — scale, flip, add, all at once

6. Set-builder notation ${\cdot ∣ \cdot}$

7. Subspace — the shape a span turns out to be

8. Determinant $det$ — the "do they fill the plane?" number

Prerequisite map

Equipment checklist

Connections