Vectors and matrices: the basic language of many AI models
From data representation to linear transforms, with a worked example and a connection to model layers.
Learning goal
Interpret vectors and matrices, perform matrix multiplication, and explain its role in a simple linear layer.
Prerequisites
- Elementary algebra
What is a vector?
A vector is both an ordered collection of numbers and a geometric object. In machine learning, vectors often represent samples, features, or embeddings.
What does a matrix do?
A matrix applies a linear transformation to a vector. In a simple neural-network layer, a bias then shifts the result.
For W=[[2,1],[-1,3]] and x=[1,2], the product is [4,5].
Why does this matter for AI?
Projections, embeddings, and many operations inside neural networks are expressed through matrix multiplication. Before attention and backpropagation, this algebra should be understood both numerically and geometrically.
Exercises
- Compute a 2×2 matrix acting on three different vectors.
- Explain why matrix multiplication order matters.
- What happens when W=I?
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See these ideas inside a real model
The path can move from mathematical structure to implementation.