-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathChapter3_1.html
More file actions
177 lines (165 loc) · 7.39 KB
/
Copy pathChapter3_1.html
File metadata and controls
177 lines (165 loc) · 7.39 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Vector Space</title>
<link href="css/style.css" rel="stylesheet" type="text/css">
</head>
<body>
<div class="header">
<div class="main">
<div class="logo"><a href="index.html"><img src="images/logo.png" width="100px"></a></div>
<div class="nav">
<ul>
<li><a href="Course.html">Courses<span class="on"></span></a></li>
<li><a href="Quiz_1.1.1.html">Quiz<span></span></a></li>
<li><a href="notes_all.html">Forum<span></span></a></li>
<li><a href="Experiment.html">Experiment<span></span></a></li>
<li><a href="video.html">More<span></span></a></li>
</ul>
</div>
<div class="sy">
<input type="text" placeholder="search" value="search">
<button>Question</button>
</div>
<div class="tx"><a href="login.html"><img src="images/icon.png"></a></div>
</div>
</div>
<div class="main">
<div class="video">
<div class="Course">
<div class="Course_z">
<h1 class="Course_z_off">Linear Algebra</h1>
<dl>
<dt>Linear Equation</dt>
<dd><a class="Course_chapter_off" href="Chapter1_1.html">Linear Equation</a></dd>
<dd><a class="Course_chapter_off" href="Chapter1_2.html">Vector Equation</a></dd>
</dl>
<dl>
<dt>Matrix Algebra</dt>
<dd><a class="Course_chapter_off" href="Chapter2_1.html">Matrix Equation</a></dd>
<dd><a class="Course_chapter_off" href="Chapter2_2.html">Subspace & Dimension</a></dd>
</dl>
<dl>
<dt>Vector Space</dt>
<dd><a class="Course_chapter_on" href="Chapter3_1.html">Vector Space</a></dd>
<dd><a class="Course_chapter_off" href="Chapter3_2.html">Zero Space & Basis</a></dd>
</dl>
<dl>
<dt>Eigenvalues</dt>
<dd><a class="Course_chapter_off" href="Chapter4_1.html">Eigenvectors&Eigenvalues</a></dd>
<dd><a class="Course_chapter_off" href="Chapter4_2.html">Eigenfunctions</a></dd>
</dl>
</div>
<div class="Course_chapter">
<h1>Vector Space</h1>
<hr color = "#708ACE" size=1px/>
<h2>What is vector space?</h2>
<div class="Course_chapter_content">
<p>A vector space is a non-empty set V of objects called vectors, on which two operations are defined, called addition and scalar multiplication, subject to the following axioms:</p>
<ol>
<li>1. u + v = v + u;</li>
<li>2. (u + v) + w = u + (v + w);</li>
<li>3. There is a vector 0 in V such that u + 0 = 0 + u = u;</li>
<li>4. For each vector u in V, there exists a vector -u in V, such that u + (-u) = -u + u =0;</li>
<li>5. c(u + v) = cu + cv;</li>
<li>6. (c + d)u = cu + du;</li>
<li>7. c(du) = (cd)u;</li>
<li>8. 1*u = u</li>
</ol>
<p>Using only these axioms, we can prove the zero vector in axiom 4 and the negative vector in axiom 5 -u are unique. </p>
<p>The definition of V is defined in a set way, involving only length and direction, independent of the coordinate system, and addition obeys the parallelogram law:</p>
<div align="center">
<img src="images/vectorspace.png" height ="105px">
<img src="images/vectorspace2.png" height ="105px">
<img src="images/vectorspace3.png" height ="105px">
</div>
</div>
<hr color="#708ACE" size=1px/>
<h2>Subspace</h2>
<div class="Course_chapter_content">
<p>In many problems, a vector space is composed of an appropriate subset of a large vector space. In this case, the ten axioms of vector space need only satisfy the following three properties to be called the subspace H of vector space V.</p>
<ol>
<li>1. The zero vector in V is a member of H;</li>
<li>2. H is closed to vector addition: for any vector u and v in H, vector u+v belongs to H;</li>
<li>3. H is closed for vector multiplication: for any vector u and the scalar c in H, cu belongs to H.</li>
</ol>
<div align = "center">
<img src="images/subspace3.png" height ="90px" >    <img src="images/subspace4.png" height ="110px" >
<h6>A subspace of V              A subspace of R^3</h6>
</div>
<p>R^2 is not subspace of R^3,because R^2 is not a subset of R^3(each vector in R^3 contains three elements,but the vector in R^2 only contains two elements).</p>
</div>
<hr color="#708ACE" size=1px/>
<div class="fy">
<div class="fy_s">
<p>Test Yourself!</p>
<a href="Quiz_3.1.1.html"><p class="text1">Quiz Link</p></a>
</div>
<div class="fy_x">
<a href="Chapter2_2.html">Previous</a>
<a href="Chapter3_2.html">Next</a>
</div>
</div>
</div>
</div>
<div class="video_right">
<div class="video_xk">
<div class="video_top">
<ul>
<li>
<a href="threads_all.html">
<img src="images/icon1.jpg">
<p>Answer<br>Question</p>
</a>
</li>
<li>
<a href="notes_all.html">
<img src="images/icon2.jpg">
<p>Share<br>notes</p>
</a>
</li>
<li>
<a href="download_all.html">
<img src="images/icon3.jpg">
<p>Download<br>file</p>
</a>
</li>
</ul>
</div>
<div class="video_btm">
<span>Answer later</span><span style="border:none;">Draft</span>
</div>
</div>
<div class="tb">
<ul>
<li>
<a href="#">
<img src="images/icon4.jpg">
<p>Reference</p>
</a>
</li>
<li>
<a href="video.html">
<img src="images/icon5.jpg">
<p>Video</p>
</a>
</li>
<li>
<a href="video.html">
<img src="images/icon6.jpg">
<p>Further<br>Study</p>
</a>
</li>
</ul>
</div>
</div>
</div>
<div class="foot">
<a href="#">AboutUs</a>|
<a href="#">Contact Us</a> |
<a href="#">Help Centre</a> |
<a href="#">Terms of Service</a> | @2020- AlgebraGO.Inc All Rights reserved.
</div>
</body>
</html>