{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"from mpl_toolkits.mplot3d import Axes3D"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# $\\mathbb{R}^n$ and Vectors "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Every vector in the vector space has a length except $0$-vector, and the terminology of length is **norm**, the general formula of norm of a vector in $\\mathbb{R}^k$is \n",
"\n",
"$$\n",
"\\|u\\|= \\sqrt{x_1^2+x_2^2+...+x_k^2}\n",
"$$\n",
"\n",
"As an example, let's plot vectors $(4, 7)$ and $(8, 6)$ in $\\mathbb{R}^2$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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