Applications of Linear Algebra

Machine Learning

Neural networks are chains of matrix multiplications and non-linearities. Each layer applies a weight matrix. Training adjusts weights via gradient descent — the gradient is computed using vector calculus and stored as matrices.

Computer Graphics

Every 3D game applies rotation, translation, and perspective matrices to every vertex per frame. A 4×4 homogeneous transformation matrix can encode rotation + translation + scale in one product.

Principal Component Analysis (PCA)

PCA compresses high-dimensional data by projecting it onto the principal components — eigenvectors of the covariance matrix. This is the foundation of face recognition, data visualization, and noise reduction.

Linear Regression

Fitting a line to data minimizes the sum of squared residuals. The solution is the least-squares projection: x = (AᵀA)⁻¹Aᵀb where A is the design matrix.

Cryptography

The Hill cipher encrypts text by multiplying letter vectors by a key matrix. Decryption requires the matrix inverse. Modern cryptography builds on modular arithmetic and discrete logarithm problems.