From a561f2c46c5f1bcfc7ceb546f2a1eab9bee0003d Mon Sep 17 00:00:00 2001 From: Deniz Donmez Date: Mon, 15 Sep 2025 14:14:31 +0300 Subject: [PATCH] Tutorial 1 - Vectors & Matrices in numethods --- tutorials/tutorial1_vectors_matrices.ipynb | 476 +++++++++++++++++++++ 1 file changed, 476 insertions(+) create mode 100644 tutorials/tutorial1_vectors_matrices.ipynb diff --git a/tutorials/tutorial1_vectors_matrices.ipynb b/tutorials/tutorial1_vectors_matrices.ipynb new file mode 100644 index 0000000..c9e3ac2 --- /dev/null +++ b/tutorials/tutorial1_vectors_matrices.ipynb @@ -0,0 +1,476 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "a9d0ab82", + "metadata": {}, + "source": [ + "# šŸ“˜ Tutorial 1 - Vectors & Matrices in `numethods`\n", + "\n", + "---\n", + "\n", + "## 1. Introduction\n", + "\n", + "In numerical computing, **vectors** and **matrices** are the basic objects. \n", + "Almost every algorithm (linear solvers, eigenvalue problems, optimization, curve fitting, etc.) is built upon them. \n", + "\n", + "In this tutorial, we will:\n", + "- Define what vectors and matrices are.\n", + "- Review their basic operations (addition, scalar multiplication, multiplication).\n", + "- Define and compute vector and matrix **norms**.\n", + "- Show how these operations work in the `numethods` package." + ] + }, + { + "cell_type": "markdown", + "id": "b91edc5c", + "metadata": {}, + "source": [ + "## 2. Importing our package" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "2107103a", + "metadata": {}, + "outputs": [], + "source": [ + "from numethods.linalg import Vector, Matrix" + ] + }, + { + "cell_type": "markdown", + "id": "f64a4d9a", + "metadata": {}, + "source": [ + "## 3. What is a vector?\n", + "\n", + "A **vector** is an ordered collection of numbers (scalars). \n", + "We can think of a vector as a column:\n", + "\n", + "$$\n", + "v =\n", + "\\begin{bmatrix}\n", + "v_1 \\\\\n", + "v_2 \\\\\n", + "\\vdots \\\\\n", + "v_n\n", + "\\end{bmatrix}, \\quad v \\in \\mathbb{R}^n\n", + "$$\n", + "\n", + "or as a row:\n", + "\n", + "$$\n", + "v^T = \\begin{bmatrix} v_1 & v_2 & \\cdots & v_n \\end{bmatrix}.\n", + "$$\n", + "\n", + "### Example\n", + "A vector in $\\mathbb{R}^3$:\n", + "\n", + "$$\n", + "v = \\begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "46c96e42", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "v = Vector([1.0, 2.0, 3.0])\n" + ] + } + ], + "source": [ + "v = Vector([1, 2, 3])\n", + "print(\"v =\", v)" + ] + }, + { + "cell_type": "markdown", + "id": "5e951a1b", + "metadata": {}, + "source": [ + "## 4. What is a matrix?\n", + "\n", + "A **matrix** is a rectangular array of numbers with rows and columns:\n", + "\n", + "$$\n", + "A =\n", + "\\begin{bmatrix}\n", + "a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n", + "a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n", + "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", + "a_{m1} & a_{m2} & \\cdots & a_{mn}\n", + "\\end{bmatrix}, \\quad A \\in \\mathbb{R}^{m \\times n}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "a092da2f", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + " Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])\n" + ] + } + ], + "source": [ + "A = Matrix([[1, 2, 3],\n", + " [4, 5, 6],\n", + " [7, 8, 9]])\n", + "print(\"A =\\n\", A)" + ] + }, + { + "cell_type": "markdown", + "id": "4f0d9ec4", + "metadata": {}, + "source": [ + "## 5. Basic operations\n", + "\n", + "### 5.1 Vector addition and subtraction\n", + "For $ u, v \\in \\mathbb{R}^n $:\n", + "\n", + "$$\n", + "u + v = \\begin{bmatrix} u_1 + v_1 \\\\ u_2 + v_2 \\\\ \\vdots \\\\ u_n + v_n \\end{bmatrix},\n", + "\\quad\n", + "u - v = \\begin{bmatrix} u_1 - v_1 \\\\ u_2 - v_2 \\\\ \\vdots \\\\ u_n - v_n \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "b7571e71", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "u + v = Vector([4.0, 4.0, 4.0])\n", + "u - v = Vector([2.0, 0.0, -2.0])\n" + ] + } + ], + "source": [ + "u = Vector([3, 2, 1])\n", + "v = Vector([1, 2, 3])\n", + "\n", + "print(\"u + v =\", u + v)\n", + "print(\"u - v =\", u - v)" + ] + }, + { + "cell_type": "markdown", + "id": "1aa8f396", + "metadata": {}, + "source": [ + "### 5.2 Scalar multiplication\n", + "For $ \\alpha \\in \\mathbb{R}, v \\in \\mathbb{R}^n $:\n", + "\n", + "$$\n", + "\\alpha v = \\begin{bmatrix} \\alpha v_1 \\\\ \\alpha v_2 \\\\ \\vdots \\\\ \\alpha v_n \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "b4168dfe", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2 * v = Vector([2.0, 4.0, 6.0])\n", + "v * 2 = Vector([2.0, 4.0, 6.0])\n" + ] + } + ], + "source": [ + "v = Vector([1, 2, 3])\n", + "\n", + "print(\"2 * v =\", 2 * v)\n", + "print(\"v * 2 =\", v * 2)" + ] + }, + { + "cell_type": "markdown", + "id": "e4919b59", + "metadata": {}, + "source": [ + "### 5.3 Matrix addition and subtraction\n", + "For $ A, B \\in \\mathbb{R}^{m \\times n} $:\n", + "\n", + "$$\n", + "A + B = [ a_{ij} + b_{ij} ], \\quad\n", + "A - B = [ a_{ij} - b_{ij} ].\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "bd544880", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A + B =\n", + " Matrix([[6.0, 8.0], [10.0, 12.0]])\n", + "A - B =\n", + " Matrix([[-4.0, -4.0], [-4.0, -4.0]])\n" + ] + } + ], + "source": [ + "A = Matrix([[1, 2], [3, 4]])\n", + "B = Matrix([[5, 6], [7, 8]])\n", + "\n", + "print(\"A + B =\\n\", A + B)\n", + "print(\"A - B =\\n\", A - B)" + ] + }, + { + "cell_type": "markdown", + "id": "0a1fb7f1", + "metadata": {}, + "source": [ + "### 5.4 Matrix-Vector multiplication\n", + "For $ A \\in \\mathbb{R}^{m \\times n}, v \\in \\mathbb{R}^n $:\n", + "\n", + "$$\n", + "(Av)_i = \\sum_{j=1}^n a_{ij} v_j.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "d65fd20e", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A @ v = Vector([-2.0, -2.0, -2.0])\n" + ] + } + ], + "source": [ + "A = Matrix([[1, 2, 3],\n", + " [4, 5, 6],\n", + " [7, 8, 9]])\n", + "v = Vector([1, 0, -1])\n", + "\n", + "print(\"A @ v =\", A @ v)" + ] + }, + { + "cell_type": "markdown", + "id": "8c2d30ec", + "metadata": {}, + "source": [ + "### 5.5 Matrix-Matrix multiplication\n", + "For $ A \\in \\mathbb{R}^{m \\times n}, B \\in \\mathbb{R}^{n \\times p} $:\n", + "\n", + "$$\n", + "(AB)_{ij} = \\sum_{k=1}^n a_{ik} b_{kj}.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "f46a96ec", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A @ B =\n", + " Matrix([[4.0, 4.0], [10.0, 8.0]])\n" + ] + } + ], + "source": [ + "A = Matrix([[1, 2],\n", + " [3, 4]])\n", + "B = Matrix([[2, 0],\n", + " [1, 2]])\n", + "\n", + "print(\"A @ B =\\n\", A @ B)" + ] + }, + { + "cell_type": "markdown", + "id": "470e38cb", + "metadata": {}, + "source": [ + "### 5.6 Transpose\n", + "For $ A \\in \\mathbb{R}^{m \\times n} $:\n", + "\n", + "$$\n", + "A^T_{ij} = A_{ji}.\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "42168bd4", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A^T =\n", + " Matrix([[1.0, 4.0], [2.0, 5.0], [3.0, 6.0]])\n" + ] + } + ], + "source": [ + "A = Matrix([[1, 2, 3],\n", + " [4, 5, 6]])\n", + "\n", + "print(\"A^T =\\n\", A.T)" + ] + }, + { + "cell_type": "markdown", + "id": "226efb8f", + "metadata": {}, + "source": [ + "## 6. Norms\n", + "\n", + "Norms measure the **size** or **length** of vectors and matrices." + ] + }, + { + "cell_type": "markdown", + "id": "120cf212", + "metadata": {}, + "source": [ + "### 6.1 Vector norms\n", + "- **1-norm**: $\\|v\\|_1 = \\sum |v_i|$ \n", + "- **2-norm (Euclidean)**: $\\|v\\|_2 = \\sqrt{\\sum v_i^2}$ \n", + "- **āˆž-norm**: $\\|v\\|_\\infty = \\max |v_i|$" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "3b187412", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ā€–vā€–ā‚ = 12.0\n", + "ā€–vā€–ā‚‚ = 7.0710678118654755\n", + "ā€–vā€–āˆž = 5.0\n" + ] + } + ], + "source": [ + "v = Vector([3, -4, 5])\n", + "\n", + "print(\"ā€–vā€–ā‚ =\", v.norm())\n", + "print(\"ā€–vā€–ā‚‚ =\", v.norm2())\n", + "print(\"ā€–vā€–āˆž =\", v.norm_inf())" + ] + }, + { + "cell_type": "markdown", + "id": "3ad8369e", + "metadata": {}, + "source": [ + "### 6.2 Matrix norms\n", + "- **Frobenius norm**: $\\|A\\|_F = \\sqrt{\\sum_{i,j} a_{ij}^2}$ \n", + "- **1-norm** (maximum column sum): $\\|A\\|_1 = \\max_j \\sum_i |a_{ij}|$ \n", + "- **āˆž-norm** (maximum row sum): $\\|A\\|_\\infty = \\max_i \\sum_j |a_{ij}|$" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "72a8ae7c", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ā€–Aā€–_F = 5.477225575051661\n", + "ā€–Aā€–ā‚ = 6.0\n", + "ā€–Aā€–āˆž = 7.0\n" + ] + } + ], + "source": [ + "A = Matrix([[1, -2],\n", + " [3, 4]])\n", + "\n", + "print(\"ā€–Aā€–_F =\", A.norm_fro())\n", + "print(\"ā€–Aā€–ā‚ =\", A.norm())\n", + "print(\"ā€–Aā€–āˆž =\", A.norm_inf())" + ] + }, + { + "cell_type": "markdown", + "id": "eab94be3", + "metadata": {}, + "source": [ + "## 7. Summary\n", + "- A **vector** is an element of $\\mathbb{R}^n$, a list of numbers. \n", + "- A **matrix** is a rectangular array of numbers $\\mathbb{R}^{m \\times n}$. \n", + "- We can add, subtract, and scale vectors/matrices. \n", + "- Multiplication extends naturally: matrix-vector and matrix*matrix. \n", + "- **Transpose** flips rows and columns. \n", + "- **Norms** measure the size of vectors and matrices." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.13.7" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +}