Linear Algebra

Linear algebra is a cornerstone of machine learning (ML), providing tools to represent and manipulate data. This section introduces vectors, matrices, and key operations, with a Rust lab using nalgebra to apply these concepts.

Vectors

A vector is an ordered list of numbers, representing a point or direction in space. In ML, vectors store features (e.g., house size, price).

Definition: A vector $v$ in $R^{n}$ is:

v = [\begin{matrix} v_{1} \\ v_{2} \\ ⋮ \\ v_{n} \end{matrix}]

where $v_{i}$ are components.

Operations:

Addition: $u + v = [u_{1} + v_{1}, u_{2} + v_{2}, \dots]$ .
Scalar Multiplication: $c v = [c v_{1}, c v_{2}, \dots]$ .
Dot Product: $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + \dots$ .

Matrices

A matrix is a rectangular array of numbers, used in ML for transformations and datasets.

Definition: A matrix $A$ with $m$ rows and $n$ columns is:

A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]

Operations:

Addition: Element-wise, for same-sized matrices.
Matrix Multiplication: For an $m \times n$ matrix $A$ and $n \times p$ matrix $B$ , $C = A B$ where $c_{i j} = \sum_{k} a_{i k} b_{k j}$ .
Transpose: $A^{T}$ swaps rows and columns.

ML Applications

Vectors: Represent data points (e.g., features in regression).
Matrices: Store datasets or model weights (e.g., neural network layers).
Operations: Used in algorithms like gradient descent and PCA.

Lab: Matrix Operations with `nalgebra`

You’ll perform vector and matrix operations using nalgebra, a Rust library for linear algebra.

Edit src/main.rs in your rust_ml_tutorial project:

rust

use nalgebra::{Matrix2, Vector2};

fn main() {
    // Define vectors
    let v1 = Vector2::new(1.0, 2.0);
    let v2 = Vector2::new(3.0, 4.0);
    let dot_product = v1.dot(&v2);
    println!("Dot Product: {}", dot_product);

    // Define matrices
    let m1 = Matrix2::new(1.0, 2.0, 3.0, 4.0);
    let m2 = Matrix2::new(5.0, 6.0, 7.0, 8.0);
    let product = m1 * m2;
    println!("Matrix Product:\n{}", product);
}

Ensure Dependencies:
- Verify Cargo.toml includes:
  toml
```
[dependencies]
nalgebra = "0.33.2"
```
- Run cargo build.

Run the Program:

bash

cargo run

Expected Output:

Dot Product: 11
Matrix Product:
[[19, 22],
 [43, 50]]

Understanding the Results

Dot Product: Computes $v_{1} \cdot v_{2} = 1 \cdot 3 + 2 \cdot 4 = 11$ , useful in ML for similarity measures.
Matrix Product: Computes $M_{1} M_{2}$ , illustrating transformations in neural networks.

This lab builds skills for ML computations, linking to Core Machine Learning.

Learning from Official Resources

Deepen Rust skills with:

The Rust Programming Language (The Book): Free at doc.rust-lang.org/book.
Programming Rust: By Blandy, Orendorff, and Tindall, ideal for ML.

Next Steps

Continue to Calculus for ML’s optimization concepts, or revisit First ML Lab.

Linear Algebra ​

Vectors ​

Matrices ​

ML Applications ​

Lab: Matrix Operations with nalgebra ​

Understanding the Results ​

Learning from Official Resources ​

Next Steps ​

Further Reading ​