-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathChapter3_2.html
More file actions
185 lines (173 loc) · 8.38 KB
/
Copy pathChapter3_2.html
File metadata and controls
185 lines (173 loc) · 8.38 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
178
179
180
181
182
183
184
185
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Zero Space, Column Space, Basis</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_off" href="Chapter3_1.html">Vector Space</a></dd>
<dd><a class="Course_chapter_on" 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>Zero Space, Column Space, Basis</h1>
<hr color = "#708ACE" size=1px/>
<h2>The null space of the matrix</h2>
<div class="Course_chapter_content">
<p>Consider the following homogeneous equations:</p>
<div align="center">
<img src="images/nullspace.png" height= "55px">
</div>
<p>In the matrix form, the system of equations can be written as Ax=0, where <img src="images/nullspace2.png" height="40px">,
and the set of all x's that satisfy the homogeneous system is called the solution set of the system. It is usually convenient to directly associate the solution set with the matrix A. We call the set of all x's that satisfy Ax=0 the null space of the matrix A.</p>
<p>The null space of the matrix is written as Null A, Null A={x: x ∈ Rn, Ax=0}, Null A is further described as the set of all vectors x in Rn mapped to the zero vector in Rn by the linear transformation x→Ax.</p>
<div align="center">
<img src="images/nullspace3.png" height= "120px">
</div>
</div>
<hr color="#708ACE" size=1px/>
<h2>The column space of the matrix</h2>
<div class="Course_chapter_content">
<p>A is an m by n matrix, and the column space is A set of all the linear combinations of the columns of A, called Col A. If A= [a1... an], Col A=Span{a1... The an}.</p>
<p>The column space of m×n matrix A is A word space of Rn, <img src="images/formula.png" height= "16px">, indicating that Col A is the range of linear transformation x→Ax.</p>
</div>
<hr color="#708ACE" size=1px/>
<h2>Linear independent</h2>
<div class="Course_chapter_content">
<p>A set of indices in V vector space {v1... Vp}, if the vector equation c1*v1+... + cp*vp =0 only trivial solution is c1=0... If cp=0, then {v1... Vp is linearly independent.</p>
<p>If the vector equation c1*v1+... + cp*vp =0 exists a non-trivial solution namely that c1... cp are not all 0, then {v1... Vp} is linearly dependent.</p>
</div>
<hr color="#708ACE" size=1px/>
<h2>Basis</h2>
<div class="Course_chapter_content">
<p>Let H be a subspace of the vector space V, the index set of the vectors in VB= {b1... <b></b>p} is called a basis for H if B is a linearly independent set.</p>
<p>The subspace generated by B is the same as H, that is, H= Span {b1... bp}.</p>
<p>Two observations of the basis:</p>
<ol>
<li>1. When using the spanning set theorem, the deletion of vectors from the spanning set must stop when the set becomes linearly independent. If one more vector is deleted, the vector will not be a linear combination of the remaining vectors, so V can no longer be generated.</li>
<li>2. The basis is still the largest linearly independent set possible. If S is a basis for V and a new vector is added to S, such as the vector V extracted from V, then the new set is no longer linearly independent, because S generates V, so w is a linear combination of the elements in S.</li>
</ol>
<div align="center">
<img src="images/formula1.png" height="120px">
<h6>linear independent, but can’t generate R^3</h6>
<img src="images/formula2.png" height="120px">
<h6>a basis of R^3</h6>
<img src="images/formula3.png" height="120px">
<h6>can generate R^3, but linear dependent</h6>
</div>
</div>
<hr color="#708ACE" size=1px/>
<div class="fy">
<div class="fy_s">
<p>Test Yourself!</p>
<a href="Quiz_3.2.1.html"><p class="text1">Quiz Link</p></a>
</div>
<div class="fy_x">
<a href="Chapter3_1.html">Previous</a>
<a href="Chapter4_1.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>