vector-spaces.html (2583B)
1 <!DOCTYPE html> 2 <html> 3 <head> 4 <script type="text/javascript" async src="https://cdn.jsdelivr.net/gh/mathjax/MathJax@2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> 5 <link rel="Stylesheet" type="text/css" href="style.css"> 6 <title>vector-spaces</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Vector spaces"><h2 id="Vector spaces">Vector spaces</h2></div> 12 <p> 13 subspace of \(\Re^n\) is any set H in \(\Re^n\) that has properties: 14 </p> 15 <ol> 16 <li> 17 The zero vector is in H 18 19 <li> 20 For each u and v in H, the sum \(u + v\) is in H 21 22 <li> 23 For each u in H and each scalar c, the vector \(cu\) is in H 24 25 </ol> 26 27 <p> 28 the zero subspace is the set that only contains zero vector in \(\Re^n\) 29 </p> 30 31 <div id="Vector spaces-Column space and null space of a matrix"><h3 id="Column space and null space of a matrix">Column space and null space of a matrix</h3></div> 32 <p> 33 column space: set of all linear combinations of the columns of a matrix. it's the <em>span</em> of the columns of the matrix. 34 </p> 35 36 <p> 37 column space of m × n matrix is subspace of \(\Re^m\) 38 </p> 39 40 <p> 41 null space: set of all solutions of equation \(Ax = 0\). 42 </p> 43 44 <p> 45 null space of an m × n matrix is subspace of \(\Re^n\). 46 </p> 47 48 <p> 49 to determine if p is in Nul A, check if Ap = 0. if so, p is in Nul A. 50 </p> 51 52 <div id="Vector spaces-Basis for a subspace"><h3 id="Basis for a subspace">Basis for a subspace</h3></div> 53 <p> 54 basis for subspace H of \(\Re^n\) is linearly independent set in H spanning H 55 </p> 56 57 <p> 58 the pivot columns of a matrix form the basis for its column space. 59 </p> 60 61 <div id="Vector spaces-Coordinates"><h3 id="Coordinates">Coordinates</h3></div> 62 <p> 63 let \(H \in \Re^n\) be subspace with \(B = \{ b_1, \dots, b_p\}\). then for all x ∈ H, there are unique \(c_1, \dots, c_p\) such that \(x = c_1 b_2 + \dots + c_p b_p\). (to prove this theorem, use a contradiction on uniqueness) 64 </p> 65 66 <p> 67 the coordinates of x w.r.t. B are \(c_1, \dots, c_p\). 68 </p> 69 70 <p> 71 the coordinate system of x w.r.t. B is \([x]_B = \begin{bmatrix}c_1\\ \dots\\ c_p\end{bmatrix}\) 72 </p> 73 74 <div id="Vector spaces-Dimension of a subspace"><h3 id="Dimension of a subspace">Dimension of a subspace</h3></div> 75 <p> 76 let \(H \in \Re^n\) be a subspace with basis \(B=\{ b_1, \dots, b_p \}\). then every basis for H comprises p vectors. 77 </p> 78 79 <p> 80 the dimension of H is the number of basis vectors in <em>any</em> basis for H. 81 </p> 82 83 <p> 84 dim Col A = #pivot columns (rank A) 85 </p> 86 87 <p> 88 dim Nul A = #free variables in Ax = 0 89 </p> 90 91 <p> 92 Rank theorem: dim Col A + dim Nul A = #columns 93 </p> 94 95 </body> 96 </html>