Upload basic linear algebra example

Author: XuhuaHuang

src/linalg_basics/Cargo.toml

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+[package]
+name = "linalg_basics"
+version = "0.1.0"
+edition = "2024"
+
+[dependencies]
+nalgebra = "0.33.2"

src/linalg_basics/src/main.rs

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+/**
+ * @file: main.rs
+ * @brief:  This project demonstrates basic linear algebra operations
+ * @author: Xuhua Huang
+ * @date: April 18, 2025
+ * @version: 1.0
+ */
+use nalgebra::{DMatrix, LU, Vector2, Vector3};
+
+fn main() {
+    let v = Vector2::new(3.0, 4.0);
+    let unit_v = v.normalize();
+
+    println!("Original: {:?}", v);
+    println!("Normalized: {:?}", unit_v);
+    println!("Length of normalized vector: {}", unit_v.norm()); // Will print 1.0
+    println!("Length of original vector: {}", v.norm()); // Will print 5.0
+
+    // Define two 3D vectors
+    let v1 = Vector3::new(1.0, 2.0, 3.0);
+    let v2 = Vector3::new(4.0, 5.0, 6.0);
+
+    // Vector addition
+    let sum = v1 + v2;
+
+    // Vector subtraction
+    let diff = v1 - v2;
+
+    // Scalar multiplication
+    let scaled = 2.0 * v1;
+
+    // Dot product
+    let dot = v1.dot(&v2);
+
+    // Cross product
+    let cross = v1.cross(&v2);
+
+    // Magnitude (Euclidean norm)
+    let magnitude = v1.norm();
+
+    // Normalization (unit vector)
+    let normalized = v1.normalize();
+
+    // Print results
+    println!("Sum: {}", sum);
+    println!("Difference: {}", diff);
+    println!("Scaled: {}", scaled);
+    println!("Dot product: {}", dot);
+    println!("Cross product: {}", cross);
+    println!("Magnitude: {}", magnitude);
+    println!("Normalized: {}", normalized);
+
+    // Create a dynamic 3x3 matrix
+    let m = DMatrix::from_row_slice(3, 3, &[2.0, -1.0, 0.0, -1.0, 2.0, -1.0, 0.0, -1.0, 2.0]);
+
+    // Perform LU decomposition
+    let lu = LU::new(m.clone());
+
+    // Solve a linear system Ax = b
+    let b = DMatrix::from_column_slice(3, 1, &[1.0, 0.0, 1.0]);
+    let x = lu.solve(&b).expect("Matrix is singular");
+
+    println!("Solution x:\n{}", x);
+
+    // Verify A * x == b
+    println!("A * x = \n{}", &m * &x);
+}