Linear Algebra Throughout the AI Pipeline
You have now learned vectors, matrices, dot products, eigenvalues, and tensors. These are not isolated topics. They form a continuous chain of operations that every AI system uses from start to finish. This final lesson shows how each concept connects to every stage of the AI pipeline, from raw data to trained model to prediction.
The AI Pipeline at a Glance
Every machine learning system follows a common sequence: prepare data, encode it numerically, pass it through a model, compute a loss, and update the model. Linear algebra is present at every single step.
Raw Data --> Vectors --> Matrix Operations --> Predictions --> Loss --> Gradients --> Update
(encode) (embed) (forward pass) (dot products) (reduce) (tensors) (matrices)
Data Preparation: Features as Vectors, Datasets as Matrices
In Module 1, you learned that every data sample becomes a vector of features. When you collect many samples, those vectors stack into a matrix where each row is a sample and each column is a feature.
Dataset matrix X: shape (1000, 20)
-- 1000 samples, each with 20 features
This is where linear algebra begins. Before any model sees your data, it is already organized as the vectors and matrices you studied in Modules 1 and 2.
Embeddings: Mapping Data to Dense Vectors
Raw categories, words, or pixels are converted into dense vectors through embedding layers. A word embedding table is a matrix of shape (vocabulary_size, embedding_dim). Looking up a word is selecting a row from this matrix.
Embedding matrix E: shape (50000, 768)
Token "hello" at index 4523 --> E[4523] --> vector of 768 numbers
This is the bridge between symbolic data and the numerical world of linear algebra, exactly what Module 1 prepared you for.
Forward Pass: Matrix Multiplications Through Layers
The core computation of a neural network is repeated matrix multiplication. Each layer multiplies its input by a weight matrix and adds a bias, as covered in Modules 2 and 3.
Layer computation: output = input @ W + b
input: shape (32, 768) -- batch of 32 embeddings
W: shape (768, 256) -- weight matrix
b: shape (256,) -- bias vector (broadcast)
output: shape (32, 256) -- transformed representations
A deep network chains many such multiplications. The weight matrices contain everything the model has learned. This is what you studied in Module 3 on matrix multiplication.
Attention: Dot Products Measuring Similarity
In transformers, the attention mechanism uses dot products to determine how much each token should attend to every other token. This is Module 4 in action.
Attention scores = Q @ K^T / sqrt(d_k)
Q (queries): shape (batch, heads, seq_len, d_k)
K (keys): shape (batch, heads, seq_len, d_k)
Scores: shape (batch, heads, seq_len, seq_len)
Each score is a dot product measuring similarity between a query and a key. Higher scores mean greater relevance. Cosine similarity, which you studied alongside dot products, is the same core idea normalized by vector magnitudes.
Dimensionality Reduction: Eigendecomposition for Feature Selection
Before training or for analysis after training, PCA and eigendecomposition (Module 5) reduce high-dimensional data to its most important directions.
Original data: shape (10000, 500) -- 500 features
After PCA: shape (10000, 50) -- 50 principal components
The eigenvectors of the covariance matrix identify the directions of greatest variance. Projecting data onto these directions compresses it while preserving the most important structure.
Training: Gradient Tensors and Weight Updates
During training, the loss with respect to every weight is computed as a gradient tensor. These gradients have the exact same shape as the weight tensors they correspond to.
Weight W: shape (768, 256)
Gradient dW: shape (768, 256)
Update: W = W - learning_rate * dW
The weight update is an element-wise tensor operation: subtract the scaled gradient from the weight. This happens for millions or billions of parameters simultaneously, all organized as tensors, the topic of Module 6.
The Complete Picture
A single prediction in a transformer model flows through every concept in this course.
1. Input text is tokenized and embedded --> Vectors (Module 1)
2. Embeddings pass through linear layers --> Matrix multiplication (Modules 2-3)
3. Attention computes token relationships --> Dot products and similarity (Module 4)
4. Representations may be analyzed with PCA --> Eigenvalues and eigenvectors (Module 5)
5. All data flows as multi-dimensional arrays --> Tensors (Module 6)
Module-to-Pipeline Mapping
| Module | Topic | Where It Appears in AI |
|---|---|---|
| 1. Vectors | Data as ordered lists of numbers | Feature vectors, word embeddings, input encoding |
| 2. Matrices | Grids of numbers, weight storage | Weight matrices, dataset batches, transformations |
| 3. Matrix Multiplication | Combining inputs with weights | Every neural network layer, forward and backward pass |
| 4. Dot Products and Similarity | Measuring alignment between vectors | Attention scores, cosine similarity, retrieval systems |
| 5. Eigenvalues and Eigenvectors | Finding principal directions | PCA, dimensionality reduction, spectral analysis |
| 6. Tensors | Multi-dimensional data containers | Batched inputs, gradient storage, multi-head attention |
Where to Go Next
Linear algebra gives you the structural foundation of AI, but it is only half the mathematical story. Probability and Statistics for AI is the companion course that covers the other half: how AI models handle uncertainty, learn from data distributions, and make probabilistic predictions. Together, these two courses provide the complete mathematical toolkit for understanding modern machine learning.
Topics you will encounter in probability and statistics include Bayes' theorem for updating beliefs, probability distributions for modeling data, maximum likelihood estimation for training models, and information theory for understanding what models learn.
Summary
- Data preparation organizes raw data into vectors and matrices (Modules 1-2)
- Embeddings map symbolic data to dense vectors using embedding matrices (Module 1)
- Forward pass chains matrix multiplications through layers (Modules 2-3)
- Attention uses dot products to compute similarity between representations (Module 4)
- Dimensionality reduction applies eigendecomposition to find principal directions (Module 5)
- Training updates weight tensors using gradient tensors of the same shape (Module 6)
- Every concept in this course appears in a real AI system -- they are not isolated topics but connected stages of one pipeline
- Probability and Statistics for AI is the recommended next course to complete your mathematical foundation
You have now built a complete understanding of the linear algebra that powers artificial intelligence. From single numbers to multi-dimensional tensors, from vector addition to eigendecomposition, every concept you have studied is actively at work in the AI systems shaping our world. The mathematics is not abstract theory -- it is the engine running behind every prediction, every recommendation, and every generated response. Take this foundation forward and keep building.