Editing Linear Algebra (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Linear Algebra is about '''Transforming Space'''.

'''1. The Vector View''': A vector $[3, 2]$ is a point in space. Linear algebra is the study of how to move, rotate, and stretch these points.

'''2. The Matrix as a Function''': A matrix is a "Machine." You put a vector in, and a new vector comes out.
* A specific matrix might '''Rotate''' everything 90 degrees.
* Another might '''Scale''' everything to be 2x bigger.
* '''Matrix Multiplication''' is just "Stacking" these machines. Rotating ''then'' Scaling is one combined matrix.

'''3. Eigenvectors (The Soul of the Matrix)''':
When you transform space, most vectors get knocked around and change direction. But for every transformation, there are special vectors—'''Eigenvectors'''—that stay pointing the same way. They are the "Axes of Rotation."
* '''Google PageRank''': Google treats the internet as a massive matrix. The "Most important websites" are actually the '''Eigenvectors''' of that matrix.

'''High-Dimensional Space''': We can only visualize 3D, but math doesn't care. Linear algebra works the same for a 1,000,000-dimensional vector. This is how AI works—a word is represented as a vector of 1536 numbers (Embeddings), and AI "calculates" the distance between words using the '''Dot Product'''.
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