Linear Combinations, Span & Basis Vectors
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Essence of Linear Algebra · Chapter 2 🔗 https://youtu.be/k7RM-ot2NWY
🧠 Big Idea
Everything in linear algebra flows from one surprisingly important operation: scaling vectors and adding them together. The concepts of basis, linear combinations, span, and linear dependence are all just different ways of looking at that single idea.
🎯 Rethinking Coordinates
Coordinates aren't just numbers — they are scalars acting on basis vectors.
In the standard xy-plane there are two special vectors:
- î (i-hat) — length 1, pointing right along x
- ĵ (j-hat) — length 1, pointing up along y
The vector [3, -2] is really: 3·î + (-2)·ĵ
The coordinates tell you how much to scale each basis vector. The basis vectors are what those scalars act on.
You can pick any two non-parallel vectors as a basis and get a perfectly valid (just different) coordinate system.
➕ Linear Combination
Scaling two vectors by any scalars a and b, then adding:
a·v + b·w
This is a linear combination of v and w. The word "linear" comes from the fact that scaling a single vector by every real number traces out a line through the origin.
🌐 Span
The span of a set of vectors = all possible vectors reachable via linear combinations of those vectors.
| Situation | Span |
|---|---|
| Two non-parallel 2D vectors | The entire 2D plane |
| Two parallel vectors (or one = zero) | A single line through the origin |
| Both are the zero vector | Just the origin (a point) |
| Two non-parallel 3D vectors | A flat sheet (plane) through the origin |
| Three 3D vectors, none in the others' span | All of 3D space |
| Third 3D vector lies in span of first two | Same flat sheet — no new reach |
Mental image: Imagine two knobs (your two scalars). Turn them freely — the tip of the resulting vector sweeps out the span.
📍 Vectors vs. Points
When thinking about one vector → picture it as an arrow. When thinking about a collection of vectors (like an entire span) → picture each as a point at its tip. The collection becomes a line, plane, or space — much less cluttered.
🔗 Linear Dependence vs. Independence
Linearly dependent — at least one vector in the set is redundant: it can be expressed as a linear combination of the others, so removing it doesn't shrink the span.
Linearly independent — every vector genuinely adds a new dimension to the span; none can be written as a combination of the others.
Simple test: if you can remove a vector without losing any reachable points, the set is dependent.
📐 Basis (Technical Definition)
A basis of a space is a set of linearly independent vectors that span that space.
- "Linearly independent" → no redundancy, each vector contributes
- "Span the space" → together they can reach every point
The standard basis {î, ĵ} satisfies both. So does any other pair of non-parallel 2D vectors.
💡 Key Takeaways
- Coordinates = scalars on basis vectors, not just raw numbers
- Linear combination = scale + add (the single core operation of linear algebra)
- Span = everything reachable; think of it as the "reach" of your vectors
- Linear dependence = one vector is already in the others' span → redundant
- Basis = a minimal, non-redundant set that spans the whole space
🔗 Series
← Ch.1 – Vectors → Ch.3 – Linear transformations and matrices