Gilbert Strang 《Introduction to Linear Algebra》 chap1 Introduction to Vectors 笔记

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Gilbert Strang Introduction to Linear Algebra chap1 Introduction to Vectors 笔记

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Introduction to Linear Algebra, Fifth Edition (2016)

Chapter 1 starts with vectors and dot products. If the class has met them before, focus quickly on linear combinations. Section 1.3 provides three independent vectors whose combinations fill all of 3-dimensional space, and three dependent vectors in a plane.

文章目录

1.1 Vectors and Linear Combinations

A brief summary

Every section opens with a brief summary to explain its contents. When you read a new section, and when you revisit a section to review and organize it in your mind, those lines are a quick guide and an aid to memory.

1 3 v + 5 w  is a typical    linear   combination   c v + d w  of the vectors  v  and  w 1\\quad 3 \\boldsymbolv+5 \\boldsymbolw\\text is a typical \\textbf linear combination c\\boldsymbolv+d \\boldsymbolw \\text of the vectors \\boldsymbolv \\text and \\boldsymbolw 13v+5w is a typical  linear combination cv+dw of the vectors v and w

2 For  v = [ 1 1 ]  and  w = [ 2 3 ] that combination is  3 [ 1 1 ] + 5 [ 2 3 ] = [ 3 + 10 3 + 15 ] = [ 13 18 ] . 2\\quad \\textFor \\boldsymbolv=\\left[\\beginarrayl1 \\\\ 1\\endarray\\right] \\text and \\boldsymbolw=\\left[\\beginarrayl2 \\\\ 3\\endarray\\right] \\textthat combination is 3\\left[\\beginarrayl1 \\\\ 1\\endarray\\right]+5\\left[\\beginarrayl2 \\\\ 3\\endarray\\right]=\\left[\\beginarrayl3+10 \\\\ 3+15\\endarray\\right]=\\left[\\beginarrayl13 \\\\ 18\\endarray\\right]. 2For v=[11​] and w=[23​]that combination is 3[11​]+5[23​]=[3+103+15​]=[1318​].

3 The vector [ 2 3 ] = [ 2 0 ] + [ 0 3 ] goes across to  x = 2  and up to  y = 3  in the  x y  plane . 3\\quad \\textThe vector \\left[\\beginarrayl2 \\\\ 3\\endarray\\right]=\\left[\\beginarrayl2 \\\\ 0\\endarray\\right]+\\left[\\beginarrayl0 \\\\ 3\\endarray\\right] \\textgoes across to x=2 \\text and up to y=3 \\text in the x y \\text plane. 3The vector[23​]=[20​]+[03​]goes across to x=2 and up to y=3 in the xy plane.

4 The combinations  c [ 1 1 ] + d [ 2 3 ] fill the whole  x y  plane . They produce every [ x y ] . 4\\quad \\textThe combinations c\\left[\\beginarrayl1 \\\\ 1\\endarray\\right]+d\\left[\\beginarrayl2 \\\\ 3\\endarray\\right] \\textfill the whole x y \\text plane. \\textThey produce every \\left[\\beginarraylx \\\\ y\\endarray\\right]. 4The combinations c[11​]+d[23​]fill the whole xy plane.They produce every[xy​].

5 The combinations  c [ 1 1 1 ] + d [ 2 3 4 ] fill a plane in  x y z  space  . Same plane for [ 1 1 1 ] , [ 3 4 5 ] . 5\\quad \\textThe combinations c\\left[\\beginarrayl1 \\\\ 1 \\\\ 1\\endarray\\right]+d\\left[\\beginarrayl2 \\\\ 3 \\\\ 4\\endarray\\right] \\textfill a plane in x y z \\text space . \\textSame plane for \\left[\\beginarrayl1 \\\\ 1 \\\\ 1\\endarray\\right],\\left[\\beginarrayl3 \\\\ 4 \\\\ 5\\endarray\\right]. 5The combinations c ​111​ ​+d ​234​ ​fill a plane in xyz space .Same plane for ​111​ ​, ​345​ ​.

6 But c + 2 d = 1 c + 3 d = 0 c + 4 d = 0 has no solution because its right side [ 1 0 0 ] is not on that plane. 6\\quad \\textBut \\left\\\\beginarraylc+2 d=1 \\\\ c+3 d=0 \\\\ c+4 d=0\\endarray\\right. \\texthas no solution because its right side \\left[\\beginarrayl1 \\\\ 0 \\\\ 0\\endarray\\right] \\textis not on that plane. 6But⎩ ⎨ ⎧​c+2d=1c+3d=0c+4d=0​has no solution because its right side ​100​ ​is not on that plane.

向量的写法和基本概念

  • We have two separate numbers v 1 v_1 v1​ and v 2 v_2 v2​. That pair produces a t w o − d i m e n s i o n a l v e c t o r v \\boldsymboltwo-dimensional \\quad vector \\quad \\boldsymbolv two−dimensionalvectorv :
    C o l u m n v e c t o r v = [ v 1 v 2 ] v 1 =  first component of  v v 2 =  second component of  v \\boldsymbol\\quad Column\\quad vector \\quad \\boldsymbolv=\\left[\\beginarraycv_1 \\\\ v_2\\endarray\\right] \\quad \\beginarraylv_1=\\text first component of \\boldsymbolv \\\\ v_2=\\text second component of \\boldsymbolv\\endarray Columnvectorv=[v1​v2​​]v1​= first component of vv2​= second component of v​