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Tutorial 1 - Vectors & Matrices in numethods

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{
"cells": [
{
"cell_type": "markdown",
"id": "a9d0ab82",
"metadata": {},
"source": [
"# πŸ“˜ 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": [
"β€–v‖₁ = 12.0\n",
"β€–vβ€–β‚‚ = 7.0710678118654755\n",
"β€–vβ€–βˆž = 5.0\n"
]
}
],
"source": [
"v = Vector([3, -4, 5])\n",
"\n",
"print(\"β€–v‖₁ =\", v.norm())\n",
"print(\"β€–vβ€–β‚‚ =\", v.norm2())\n",
"print(\"β€–vβ€–βˆž =\", 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": [
"β€–Aβ€–_F = 5.477225575051661\n",
"β€–A‖₁ = 6.0\n",
"β€–Aβ€–βˆž = 7.0\n"
]
}
],
"source": [
"A = Matrix([[1, -2],\n",
" [3, 4]])\n",
"\n",
"print(\"β€–Aβ€–_F =\", A.norm_fro())\n",
"print(\"β€–A‖₁ =\", A.norm())\n",
"print(\"β€–Aβ€–βˆž =\", 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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