Linear Combinations, Span & Basis Vectors

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.

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