{
"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\n",
"import scipy as sp\n",
"import scipy.linalg\n",
"import sympy as sy\n",
"sy.init_printing() \n",
"# import warnings\n",
"# warnings.filterwarnings(\"ignore\")"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import linAlgVis"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Basis "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Basis**, denoted as $B$, is the minimum unit of user-customization of coordinates, which is any type of coordinate system other than Cartesian."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Basis of $\\mathbb{R}^2$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Formally speaking, the basis is a set of vectors $B$ in vector space $V$ with two conditions:\n",
"1. All vectors in $B$ are independent. \n",
"2. $\\text{span}(B)=V$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have seen _standard basis_ in earlier chapters, for instance in $\\mathbb{R}^2$\n",
"\n",
"$$e_1=\n",
"\\left[\n",
"\\begin{matrix}\n",
"1\\\\0\n",
"\\end{matrix}\n",
"\\right],\n",
"\\ e_2=\\left[\n",
"\\begin{matrix}\n",
"0\\\\1\n",
"\\end{matrix}\n",
"\\right]\n",
"$$\n",
"\n",
"and in $\\mathbb{R}^3$\n",
"\n",
"$$e_1=\n",
"\\left[\n",
"\\begin{matrix}\n",
"1\\\\0\\\\0\n",
"\\end{matrix}\n",
"\\right],\n",
"\\ e_2=\\left[\n",
"\\begin{matrix}\n",
"0\\\\1\\\\0\n",
"\\end{matrix}\n",
"\\right],\n",
"\\ e_3=\\left[\n",
"\\begin{matrix}\n",
"0\\\\0\\\\1\n",
"\\end{matrix}\n",
"\\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we show the linear combination of standard basis for vector $(3, 6)$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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