Linear Algebra Intuition
Building geometric intuition for vector spaces and transformations
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Linear algebra feels abstract when you first encounter it. Matrices, vectors, eigenvalues — they seem like arbitrary mathematical objects. But there's a beautiful geometric story underneath all the notation.
Vectors Are More Than Arrays
When you see [3, 2], don’t just think “two numbers in brackets.” Think of it as a vector:
- An arrow from the origin to the point (3, 2)
- A movement: 3 units right, 2 units up
- A way to encode direction and magnitude
Every vector is fundamentally about space and position.
Matrices Are Transformations
Here’s where it gets interesting. A matrix isn’t just a grid of numbers — it’s a transformation of space.
When you multiply a matrix by a vector, you’re transforming that vector. The matrix describes how every point in space moves.Think of it like Instagram filters — each filter is a transformation that changes every pixel according to some rule.
Example: A Simple Rotation
[cos(θ) -sin(θ)]
[sin(θ) cos(θ)]This matrix rotates every point in 2D space by angle θ. Beautiful, right?
The Magic of Basis Vectors
In 2D, we typically use [1, 0] and [0, 1] as our basis — they point along the x and y axes. But here’s the insight:
Every matrix tells you where the basis vectors land after transformation.
The first column shows where [1, 0] goes. The second column shows where [0, 1] goes. Everything else follows from linearity.
Why This Matters for Machine Learning
Neural networks are just layers of matrix transformations! When we train a model, we're learning the right transformations to map inputs to outputs.
Understanding this geometric view helps you:
- Visualize what your model is doing
- Understand dimensionality reduction (PCA, embeddings)
- Grasp why certain operations work
Eigenvalues and Eigenvectors: The Special Directions
Some vectors don’t change direction when transformed — they only get scaled. These are eigenvectors, and their scaling factors are eigenvalues.Google’s PageRank algorithm is essentially finding the dominant eigenvector of the web’s link matrix!
They reveal the fundamental “axes” of a transformation, showing you which directions are special or preserved.
Building Intuition
The key to understanding linear algebra isn’t memorizing formulas. It’s training yourself to:This geometric perspective is exactly how graphics cards (GPUs) think about the world — everything is transformations in space!
Once you see the geometry, the algebra becomes clear.
Resources I love: 3Blue1Brown’s “Essence of Linear Algebra” series is phenomenal for visual intuition.
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