Linear Algebra for AI

What Does Math Have to Do with Machine Learning?

All programming involves math at some level — and machine learning is no exception.
In fact, machine learning is programming by optimization, and to understand optimization, we need mathematics.

To understand what optimization is, how it works, and how machines learn, we need a strong mathematical foundation.

In this series, we will cover the first mathematical tool: Linear Algebra, which plays a major role in understanding optimization and its use in machine learning.

Linear Algebra

Linear algebra helps us understand:

The object being optimized
How data is structured
How models are built and computed

Takeaway for Today

Linear Algebra is important, and it is not the same as high-school algebra.

Why do we care about linear algebra?

Because it is the mathematics of arrays — and in machine learning, everything is made of arrays:

The data, like an image, is an array
The models are collections of arrays
Even the internal computations of these models are performed using arrays

What Is the Role of Linear Algebra in GPUs?

GPUs (Graphics Processing Units) are designed to perform many small math operations in parallel. Most of these operations are linear algebra tasks, like:

Matrix multiplication
Vector addition
Dot products

These operations are used in:

Graphics (e.g., rotating 3D objects, lighting)
AI/ML (e.g., neural networks where weights and inputs are matrices/vectors)

GPUs are perfect for this because:

They can do the same math operation on thousands of numbers at once
Linear algebra is highly parallel, making it a natural fit

What Type of Linear Algebra Is Used in GPUs?

Concept	Description
Vectors	Lists of numbers (1D)
Matrices	Tables of numbers (2D)
Matrix Multiplication	Core for transforming data and neural net calculations
Dot Product	Used in graphics and ML to calculate similarity
Transpose, Inverse	Used in transformations and solving equations

Real AI Example

In deep learning:

Inputs (like images or text) are represented as vectors/matrices
Weights are also stored as matrices
The output is computed using matrix multiplications

The GPU is the engine that performs these operations quickly.

What Is the Role of Linear Algebra in AI?

Linear algebra is the engine of AI.
It helps us represent and transform data — whether it’s images, audio, or text — into a mathematical format that machines can learn from.

AI works with numbers, and linear algebra gives us a clean way to:

Store data (as vectors and matrices)
Transform data (rotate, scale, combine)
Learn patterns (using matrix multiplications and updates)

Simple Example

A black and white image is just a grid of numbers (pixel brightness) — this is a matrix.

To process the image:

AI uses matrix operations, like multiplying it with a weight matrix
This helps the model detect edges, corners, and patterns

The same applies to text and audio — everything is turned into matrices or vectors.

Algorithms from Linear Algebra Used in AI

Algorithm / Concept	Use in AI	Example
Matrix Multiplication	Core of neural networks	Input × Weights = Output
Dot Product	Measures similarity	Word embeddings similarity
Eigenvalues/Eigenvectors	Dimensionality reduction (PCA)	Compress high-dimensional data
Singular Value Decomposition (SVD)	Recommendation systems	Netflix/movie suggestions
LU / QR Decomposition	Solving systems of equations	Optimization problems
Transpose, Inverse	Reshaping and solving systems	Reversing transformations

Comparison & Transformation

Without Linear Algebra	With Linear Algebra
Raw data handled manually	Data represented as matrices
Hard to find patterns	Easy to apply transformations
Slow calculations	Fast parallel operations on GPUs
Hard to scale to big data	Scales well using matrix operations

Summary

Linear Algebra is the core language of AI
It lets AI store, transform, and learn from data
It is used deeply in both hardware (GPU) and software (code)
Key operations like dot products, matrix multiplication, decompositions power every AI model

Hardware Perspective (GPU & CPU)

GPUs love matrices: They can process thousands of matrix multiplications at once
Faster matrix operations mean faster training
All deep learning frameworks (like TensorFlow and PyTorch) rely on GPU acceleration for this reason

Linear algebra fits perfectly into hardware like GPUs, which are built for parallel math.
GPUs contain thousands of small cores that can multiply numbers and add them — ideal for matrix operations in AI.

So when your model runs input × weights, the GPU does this fast and in parallel, unlike a CPU that works step by step.

Examples:

Training a neural network with 1 million weights may take hours on a CPU, but minutes on a GPU
Processing an image with 1,000,000 pixel values × 10,000 weights is handled much faster on a GPU

Software Perspective (AI Libraries & Code)

AI frameworks like TensorFlow, PyTorch, and NumPy are built on top of linear algebra libraries such as:

BLAS (Basic Linear Algebra Subprograms)
cuBLAS (NVIDIA’s GPU-accelerated version)

BLAS provides a standard interface, which means AI software written for it can run efficiently on many kinds of hardware.
NVIDIA cuBLAS is a GPU-accelerated library optimized for AI and High-Performance Computing (HPC). It provides drop-in support for industry-standard linear algebra operations.

When you write something like:

output = layer(input)

That code actually performs matrix multiplication in the background.
Even if you don’t see the math, it’s all vectors, matrices, and dot products under the hood.

Example:
Word2Vec converts words into vectors and finds similarity using the dot product — powered entirely by linear algebra.

Conclusion

Linear Algebra is not optional — it's essential for understanding how machine learning models are built, trained, and deployed.
From the way we store data, to how we train models, and the hardware/software we use — linear algebra is at the center of it all.

By learning it, you're not just learning math — you're learning the language of AI.

P.S:if you spot any mistakes, feel free to point them out — we’re all here to learn together! 😊

Haris
FAST-NUCES
BS Computer Science | Class of 2027

🔗 Portfolio: zenvila.github.io

🔗 GitHub: github.com/Zenvila

🔗 LinkedIn: linkedin.com/in/haris-shahzad-7b8746291
🔬 Member: COLAB (Research Lab)