# Learn linear algebra — the mathematics powering machine learning and graphics

Vectors, matrices, and transformations explained from first principles

Linear algebra is the branch of mathematics that deals with vectors, matrices, and linear transformations. It is the mathematical language of machine learning, computer graphics, physics simulations, and data science. When you hear about neural networks, PCA, or 3D rendering, linear algebra is what's happening under the hood.

The subject has a reputation for being abstract, but the core ideas are deeply geometric — vectors are arrows in space, matrix multiplication is a transformation, and eigenvalues describe how that transformation stretches or rotates things. Once you see the geometry, the algebra becomes intuitive rather than mysterious.

Planana's AI tutor builds your linear algebra understanding in a logical sequence, connecting the abstract notation to concrete visual intuitions at each step.

## What you'll learn

### Vectors & Vector Operations

What vectors are geometrically and algebraically — addition, scalar multiplication, dot product, and cross product.

### Matrices & Matrix Multiplication

Matrices as linear transformations, how multiplication composes transformations, and why order matters.

### Systems of Linear Equations

Solving Ax = b using Gaussian elimination and understanding when solutions exist, are unique, or are infinite.

### Determinants

What the determinant measures geometrically (volume scaling), and its role in invertibility and solving linear systems.

### Eigenvalues & Eigenvectors

The directions a transformation doesn't rotate — critical for understanding PCA, PageRank, and stability analysis.

### Vector Spaces & Subspaces

The abstract framework unifying all of linear algebra: spans, bases, dimension, and rank.

### Matrix Decompositions

LU, QR, and SVD decompositions — the workhorses of numerical computation and dimensionality reduction.

## Why learn Linear Algebra?

Linear algebra is a hard prerequisite for machine learning and deep learning. You cannot deeply understand how neural networks train, why attention mechanisms work, or what PCA is doing without a solid grasp of matrices, eigenvectors, and vector spaces.

It also appears in computer graphics, quantum computing, optimization, and signal processing. For anyone pursuing a technical career in AI, data science, or engineering, linear algebra is one of the highest-return mathematical investments you can make.

## Start learning Linear Algebra today

Tell Planana's AI tutor your goal and get a personalized daily study plan — free.