Overview

Linear Algebra is the study of vectors, vector spaces, and linear transformations. It is the mathematics of “flat” spaces (lines, planes, hyperplanes) and is fundamental to almost all areas of modern mathematics and science.

Core Idea

The core idea is linearity: properties that scale and add nicely. If you double the input, you double the output ($f(cx) = cf(x)$). If you add two inputs, you add the outputs ($f(x+y) = f(x) + f(y)$).

Formal Definition

Linear algebra studies vector spaces over a field (like the real numbers) and linear maps between these spaces. Key structures include vectors, matrices, and systems of linear equations.

Intuition

Think of it as the spreadsheet of math. A matrix is just a grid of numbers. Linear algebra gives us the rules for manipulating these grids to solve huge systems of equations simultaneously, rotate objects in 3D space, or compress images.

Examples

  • Solving Equations: finding $x, y, z$ where $2x + y = 5$ and $x - z = 2$.
  • Computer Graphics: Rotating a character on screen involves multiplying their coordinate vectors by a rotation matrix.
  • Google PageRank: The algorithm that ranks web pages is essentially finding the eigenvector of a massive matrix representing the web.

Common Misconceptions

  • Misconception: It’s just about solving systems of equations.
    • Correction: While that’s a key application, the abstract concepts of vector spaces and linear transformations are much broader and apply to functions, signals, and quantum states.
  • Misconception: It’s limited to straight lines.
    • Correction: “Linear” refers to the operations (addition/scaling), not just the geometry. You can have a vector space of polynomials (which are curved).
  • Vector Space: A collection of objects (vectors) that can be added together and multiplied by scalars.
  • Matrix: A rectangular array of numbers.
  • Eigenvalue: A scalar associated with a linear system of equations.

Applications

  • Machine Learning: Neural networks are basically layers of linear algebra operations (matrix multiplications).
  • Quantum Mechanics: States are vectors in a Hilbert space; observables are linear operators.
  • Engineering: Used to model stress, heat, fluid flow, and circuits.

Criticism and Limitations

  • Non-Linearity: Most real-world phenomena are non-linear (chaos, turbulence). Linear algebra is often just a local approximation (linearization) of complex reality.

Further Reading

  • Linear Algebra Done Right by Sheldon Axler
  • Introduction to Linear Algebra by Gilbert Strang