{
 "cells": [
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    "# 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",
    "- **∞-norm**: $\\|v\\|_\\infty = \\max |v_i|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3b187412",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "‖v‖₁ = 12.0\n",
      "‖v‖₂ = 7.0710678118654755\n",
      "‖v‖∞ = 5.0\n"
     ]
    }
   ],
   "source": [
    "v = Vector([3, -4, 5])\n",
    "\n",
    "print(\"‖v‖₁ =\", v.norm())\n",
    "print(\"‖v‖₂ =\", v.norm2())\n",
    "print(\"‖v‖∞ =\", 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",
    "- **∞-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": [
      "‖A‖_F = 5.477225575051661\n",
      "‖A‖₁ = 6.0\n",
      "‖A‖∞ = 7.0\n"
     ]
    }
   ],
   "source": [
    "A = Matrix([[1, -2],\n",
    "            [3,  4]])\n",
    "\n",
    "print(\"‖A‖_F =\", A.norm_fro())\n",
    "print(\"‖A‖₁ =\", A.norm())\n",
    "print(\"‖A‖∞ =\", 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."
   ]
  }
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