Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Every coordinate you've ever plotted was a set of instructions you never read. Two numbers... not a location, but a recipe. Scale this. Scale that. Add them together. BAM, you arrive somewhere. That's linear algebra whispering its secret in plain sight.

The Ingredients Were Always There

Think about this. When someone hands you coordinates like (3, -2), most of us just plot the point and move on. But Grant Sanderson over at 3Blue1Brown wants you to slow down and see what's really happening.

Those numbers aren't addresses. They're multipliers.

The 3 stretches a tiny unit vector called basis vectors|î (pointing right) by a factor of three. The -2 flips and stretches another unit vector called basis vectors|ĵ (pointing up) by two. Add those scaled arrows together... and only then do you arrive at your vector.

Two scaled vectors. One sum. That's called a linear combination. And it's the beating heart of everything that follows.

Choosing Your Foundation Changes Everything

Here's where it gets beautiful.

î and ĵ aren't the only basis vectors you could choose. They're just the default... the ones your textbook handed you without asking permission. You could pick two completely different arrows... one pointing up and to the right, another pointing down and to the right... and build an entirely new coordinate systems|coordinate system.

Same space. Different language.

It's like this... imagine you're building a world in Minecraft. The blocks you choose to build with determine what structures are possible. Swap out your materials and the same blueprint produces something unrecognizable. Your basis vectors are those building blocks. Change them, and the meaning of every pair of numbers shifts underneath you.

Every time we describe a vector with numbers, we're making an implicit choice about which basis we're using. Most people never notice. Now you will.

What Can You Reach?

So you've got two vectors and the freedom to scale them however you want. The natural question... what's reachable?

The answer has a name: span.

The span of two vectors is the set of every possible point you can land on by scaling those vectors and adding them together. For most pairs of 2D vectors, the span is... everything. The entire flat, infinite plane. Every point. Every direction. All of it within your grasp.

But sometimes your two vectors line up. Point the same direction. And suddenly your span collapses from a plane to a single line through the origin. Like owning two lightsabers that only swing in the same arc... looks impressive, reaches nowhere new. 🎯

This is where the visual trick matters. Forget the arrows for a second. When you're thinking about a whole collection of vectors... hundreds, thousands, infinite numbers of them... stop imagining arrows. Just see the points at their tips. A line of dots. A plane of dots. A volume of dots. The arrows served their purpose. Now let the geometry breathe.

The Third Dimension and the Question of Redundancy

Take two vectors in three-dimensional space that don't line up. Their span forms a flat sheet... a plane slicing through the origin. Beautiful. But flat.

Now add a third vector.

If that third vector is already sitting on the plane... already reachable by some combination of the first two... nothing changes. It's linear dependence|linearly dependent. Redundant. Like adding a third Fellowship member who only knows the same path Frodo and Sam already walk. Brave, sure. But not expanding the map.

However. If that third vector points off the plane... somewhere the first two couldn't reach on their own... everything opens up. That flat sheet sweeps through all of 3D space like a door swinging on a hinge. You now have access to every point in the volume. That vector is linear independence|linearly independent. It earned its place.

This is the difference between carrying dead weight and unlocking new territory.

The Definition That Earns Its Name

A basis of a space is a set of linear independence|linearly independent vectors that span the full space.

Read that again.

Linearly independent... meaning none of them are redundant. Each one opens a direction the others can't reach. And together, they span everything. Every point. Every corner. Full coverage, zero waste.

That's not just elegant math. That's a life principle hiding inside an equation.

Think about the people around you... your team, your family, your crew. Are they linearly independent? Does each person bring a dimension the others can't? Or are some just echoing directions already covered... not because they lack value, but because they haven't found the axis where their contribution is irreplaceable?

Why This Matters Beyond the Textbook

Linear algebra isn't about crunching numbers in a vacuum. It's about understanding what's reachable given the tools you start with. What space can you fill? What dimensions are you missing? Where is the redundancy you haven't noticed?

Those questions apply to vectors. They also apply to how we build teams, design curricula, allocate resources, and structure our own lives. The WHELHO Wheel has eight sections for a reason... Spirit, Mind, Body, Relationships, Money, Recreation, Work, Charity. Each one is a basis vector of a well-lived life. Collapse two of them together and your span shrinks. Keep them independent and you cover the full space of what it means to be human. 💙

The math is teaching us something if we're willing to listen.

Light doesn't fight with darkness... it just shows up. And sometimes it shows up as a coordinate system you didn't know you were free to choose.

So here's your puzzle, youngling. Look at the foundations you're standing on... the assumptions, the defaults, the basis vectors someone else handed you. Are they serving the space you're trying to reach? Or is it time to choose new ones? The math says you can. The span of your life depends on it. ✨

--- Source: https://www.youtube.com/watch?v=k7RM-ot2NWY