因?yàn)樵谌W(wǎng)搜索Gilbert Strang編寫(xiě)的Introduction to Linear Algebra時(shí)找不到中文翻譯版，所以有了人工翻譯這本書(shū)的想法。這本書(shū)的PDF本站有，這里就不貼了。
【書(shū)名】Introduction to Linear Algebra
【作者】Gilbert Strang
【版本】2016年發(fā)行的第五版
【開(kāi)始時(shí)間】2020年6月17日

目錄暫時(shí)不翻譯

• • Preface
  • • The Fifth Edition第五版
  • 1 Introduction to Vectors
  • • 1.1 Vectors and Linear Combinations
    • 1.2 Lengths and Dot Products
    • 1.3 Matrices
  • 3 Vector Spaces and Subspaces
  • • 3.1 Spaces of Vectors
    • • Subspaces
    • 3.2 The Nullspace of A: Solving Ax=0 and Rx=0
    • 3.3 The Complete Solution to Ax=b
    • 3.4 Independence, Basis and Dimension
    • 3.5 Dimensions of the Four Subspaces

Preface

I am happy for you to see this Fifth Edition of Introduction to Linear Algebra.This is the text for my video lectures on MIT’s OpenCourseWare(ocw. mit. edu and also YouTube).I hope those lectures will be useful to you(maybe even enjoyable!).
我很高興你們可以使用《Introduction to Linear Algebra》第五版，這是我在麻省理工學(xué)院的OpenCourseWare（** ocw.mit.edu 以及 YouTube **）上的視頻講座的課本。我希望這些課程對(duì)你們將很有用（甚至可能很有趣！）。
譯者注：看過(guò)視頻課的同學(xué)應(yīng)該很清楚，這位老師很喜歡開(kāi)玩笑或者搞怪，整本書(shū)也存在不少俏皮的話(huà)。
Hundreds of colleges and universities have chosen this textbook for their basic linear algebra course.A sabbatical gave me a chance to prepare two new chapters about probability and statistics and understanding data.Thousands of other improvements too-probably only noticed by the author…Here is a new addition for students and all readers:
數(shù)以百計(jì)的高校選擇了這本教科書(shū)作為其線(xiàn)性代數(shù)的基礎(chǔ)課程。一個(gè)假期使我有機(jī)會(huì)編寫(xiě)關(guān)于概率和統(tǒng)計(jì)學(xué)以及理解數(shù)據(jù)（？？）的兩章新內(nèi)容。然而很多修訂很可能只有作者才注意到…這是針對(duì)學(xué)生和所有讀者的新增內(nèi)容：

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.
每章節(jié)開(kāi)始會(huì)有一段摘要。當(dāng)你學(xué)習(xí)新的一章或者回顧某一章或者想在腦海中建立章結(jié)構(gòu)時(shí)，這段摘要可以作為指引及幫助記憶。

Another big change comes on this book’s website math. mit. edu/linearalgebra. That site now contains solutions to the Problem Sets in the book. With unlimited space, this is much more flexible than printing short solutions. There are three key websites:
本書(shū)的網(wǎng)站上還有另一個(gè)重大變化。該站點(diǎn)現(xiàn)在包含本書(shū)中課后習(xí)題的參考答案。因?yàn)椴皇苡∷婷娴南拗?#xff0c;現(xiàn)在可以把答案寫(xiě)的更詳盡。有三個(gè)主要網(wǎng)站：

ocw. mit. edu Messages come from thousands of students and faculty about linear algebra on this OpenCourseWare site. The 18.06 and 18.06 SC courses include video lectures of a complete semester of classes. Those lectures offer an independent review of the whole subject based on this textbook——the professor’s time stays free and the student’s time can be 2 a.m.(The reader doesn’t have to be in a class at all.) Six million viewers around the worid have seen these videos(amazing).I hope you find them helpful.
ocw. mit. edu：在這個(gè)公開(kāi)課網(wǎng)站上，收集了來(lái)著成千上萬(wàn)的學(xué)生和教職員工的有關(guān)線(xiàn)性代數(shù)的材料。 18.06和18.06 SC課程包括整個(gè)學(xué)期的視頻課程。這些視頻課提供了一種更自由的學(xué)習(xí)方式，教授的時(shí)間保持自由，學(xué)生的時(shí)間可以隨時(shí)上課（學(xué)生根本不必到教室上課）。全球有600萬(wàn)觀眾觀看了這些視頻（太棒了）。希望你能發(fā)現(xiàn)這些課程的有用之處。

web. mit. edu/18.06 This site has homeworks and exams(with solutions) for the current course as it is taught, and as far back as 1996. There are also review questions, Java demos, Teaching Codes, and short essays(and the video lectures). My goal is to make this book as useful to you as possible, with all the course material we can provide.
【這段不細(xì)翻了】
web. mit. edu/18.06：這個(gè)網(wǎng)站可以追溯到1996年，提供有關(guān)當(dāng)前課程的家庭作業(yè)和考試（包括解決方案），還包括復(fù)習(xí)題，Java演示，教學(xué)代碼和短文（以及視頻講座）。我的目標(biāo)是通過(guò)我們提供的所有課程資料，使本書(shū)對(duì)您盡可能有用。

math. mitedu/linearalgebra This has become an active website. It now has Solutions to Exercises-with space to explain ideas. There are also new exercises from many dif-ferent sourcespractice problems, development of textbook examples, codes in MATLAB and Julia and Python, plus whole collections of exams (18.06 and others) for review.Please visit this linear algebra site. Send suggestions to linearalgebrabook@ gmail. com
【這段不細(xì)翻了】
math. mitedu/linearalgebra：這已成為一個(gè)活躍的網(wǎng)站。現(xiàn)在，它具有“練習(xí)解決方案”，并帶有解釋思想的空間。還有許多來(lái)自不同來(lái)源的新練習(xí)，包括練習(xí)題，教科書(shū)示例的開(kāi)發(fā)，MATLAB，Julia和Python中的代碼以及完整的考試集（18.06及其他）供復(fù)習(xí)。請(qǐng)?jiān)L問(wèn)此線(xiàn)性代數(shù)站點(diǎn)。發(fā)送建議到linearalgebrabook @ gmail。 com

The Fifth Edition第五版

The cover shows the Four Fundamental Subspaces-the row space and nullspace are on the left side, the column space and the nullspace of AT are on the right. It is not usual to put the central ideas of the subject on display like this! When you meet those four spaces in Chapter 3, you will understand why that picture is so central to linear algebra.
封面展示四個(gè)向量(？？)——行空間和null空間在左側(cè)？？，A的轉(zhuǎn)置的列空間和null空間在右側(cè)？？。像我這樣展示整個(gè)課程的中心思想的做法并不常見(jiàn)！當(dāng)你在第3章中遇到這四個(gè)空間時(shí)，您將理解為什么該圖對(duì)于線(xiàn)性代數(shù)如此重要。

Those were named the Four Fundamental Subspaces in my first book, and they start from a matrix A. Each row of A is a vector in n-dimensional space. When the matrix has m rows, each column is a vector in m-dimensional space. The crucial operation in linear algebra is to take linear combinations of column vectors. This is exactly the result of a matrix-vector multiplication. Ax is a combination of the columns of A.
在我的第一本書(shū)中，它們被稱(chēng)為四個(gè)基本子空間？？，從矩陣A開(kāi)始介紹。A的每一行都是n維空間中的向量。當(dāng)矩陣有m行時(shí)，每一列又都是m維空間中的向量。線(xiàn)性代數(shù)中的關(guān)鍵操作是對(duì)列向量進(jìn)行線(xiàn)性組合。這正是矩陣向量乘法的結(jié)果。 Ax是對(duì)矩陣A中的列向量進(jìn)行線(xiàn)性組合。

When we take all combinations Ax of the column vectors, we get the column space. If this space includes the vector b, we can solve the equation Ax = b.
當(dāng)對(duì)于Ax的列向量，取得所有可能的取值時(shí)，我們就得到列向量空間。如果該空間包含向量b，我們可以求解方程Ax = b。

May I call special attention to Section 1.3, where these ideas come early-with two specific examples. You are not expected to catch every detail of vector spaces in one day! But you will see the first matrices in the book, and a picture of their column spaces. There is even an inverse matrix and its connection to calculus. You will be learning the language of linear algebra in the best and most efficient way: by using it.
我想特別注意第1.3節(jié)，其中有兩個(gè)具體的例子，這些想法很早就出現(xiàn)了？？。預(yù)計(jì)你不會(huì)在一天內(nèi)掌握向量空間的所有知識(shí)！但是您會(huì)看到書(shū)中的第一個(gè)矩陣，以及它們的列向量空間的圖片。甚至還有一個(gè)逆矩陣及其與微積分的聯(lián)系。您將以最佳和最有效的方式學(xué)習(xí)線(xiàn)性代數(shù)的語(yǔ)言：通過(guò)使用它。

Every section of the basic course ends with a large collection of review problems. They ask you to use the ideas in that section–the dimension of the column space, a basis for that space, the rank and inverse and determinant and eigenvalues of A. Many problems look for computations by hand on a small matrix, and they have been highly praised. The Challenge Problems go a step further, and sometimes deeper. Let me give four examples:
基礎(chǔ)課程每個(gè)部分的結(jié)尾都有大量的復(fù)習(xí)題。他們要求您使用該部分中的知識(shí)去解答問(wèn)題，這些知識(shí)包含列空間的維數(shù)，該向量空間的基，A的秩和逆以及行列式和特征值。許多問(wèn)題都需要?jiǎng)邮秩ビ?jì)算，這些問(wèn)題對(duì)于學(xué)習(xí)這門(mén)課程都十分有用。這些具有挑戰(zhàn)性的問(wèn)題設(shè)計(jì)的內(nèi)容會(huì)更深。讓我舉四個(gè)例子：

Section 2.1: Which row exchanges of a Sudoku matrix produce another Sudoku matrix?
Section 2.7: If P is a permutation matrix, why is some power p^k equal to I?
Section 3.4: If Ax= b and Cx = b have the same solutions for every b, does A equal C?
Section 4.1: What conditions on the four vectors r, n, c, ￡ allow them to be bases for the row space, the nullspace, the column space, and the left nullspace of a 2 by 2 matrix?
第2.1節(jié)：數(shù)獨(dú)中的哪些行交換之后依然是一個(gè)數(shù)獨(dú)？
第2.7節(jié)：如果P是一個(gè)置換矩陣，為什么有些冪p ^ k等于I？？？
第3.4節(jié)：如果Ax = b和Cx = b對(duì)于每個(gè)b都有相同的解，那么A是否等于C？
第4.1節(jié)：對(duì)于四個(gè)向量r，n，c，￡，當(dāng)滿(mǎn)足哪些條件時(shí)允許它們成為2 x 2矩陣的行向量空間，空向量空間？？，列向量空間和左空向量空間？？的基？

1 Introduction to Vectors

1.1 Vectors and Linear Combinations

1.2 Lengths and Dot Products

1.3 Matrices

3 Vector Spaces and Subspaces

3.1 Spaces of Vectors

1 The standard n-dimensional space Rn contains all real column vectors with n components.
1標(biāo)準(zhǔn)n維空間Rn包含具有n個(gè)分量的所有實(shí)列向量。
2 If v and w are in a vector space S, every combination cv + dw must be in S.
2如果v和w在向量空間S中，則cv + dw的每個(gè)組合都在S中。
3 The “vectors” in S can be matrices or functions of x. The 1-point space Z consists of x = 0.
3 S中的“向量”可以是x的矩陣或函數(shù)。 單點(diǎn)空間Z由x = 0組成。
4 A subspace of Rn is a vector space inside Rn. Example: The line y = 3x inside R2 .
4 Rn的子空間是Rn內(nèi)部的向量空間。示例：y = 3x包含于R2。
5 The column space of A contains all combinations of the columns of A: a subspace of Rm.
5 A的列空間包含A的列的所有組合，A的列空間是Rm的子空間。
6 The column space contains all the vectors Ax. So Ax = b is solvable when b is in C(A).
6 列空間包含所有向量Ax。因此，當(dāng)b在C（A）中時(shí)，Ax = b是可解的。

To a newcomer, matrix calculations involve a lot of numbers. To you, they involve vectors. The columns of Ax and AB are linear combinations of n vectors-the columns of A. This chapter moves from numbers and vectors to a third level of understanding (the highest level). Instead of individual columns, we look at “spaces” of vectors. Without seeing vector spaces and especially their subspaces, you haven’t understood everything about Ax= b.
對(duì)于新手來(lái)說(shuō)，矩陣計(jì)算涉及很多數(shù)字知識(shí)。對(duì)您來(lái)說(shuō)，它們涉及向量。 Ax和AB的列是n個(gè)向量（A的列）的線(xiàn)性組合。本章從數(shù)字和向量移至第三級(jí)理解（最高級(jí)別）。我們關(guān)注向量的“空間”而不是單個(gè)列。如果沒(méi)有想象到向量空間，尤其是它們的子空間，您將不了解關(guān)于Ax = b的所有知識(shí)。

Since this chapter goes a little deeper, it may seem a little harder. That is natural. We are looking inside the calculations, to find the mathematics. The author’s job is to make it clear. The chapter ends with the “Fundamental Theorem of Linear Algebra”.
由于本章的內(nèi)容會(huì)更深?yuàn)W一些，因此可能會(huì)更難一些。那是很自然的。我們正在尋找計(jì)算過(guò)程中的數(shù)學(xué)原理。作者的工作將這些原理講清楚。本章以“線(xiàn)性代數(shù)的基本定理”結(jié)尾。

We begin with the most important vector spaces. They are denoted by R^1 , R^2, R^3, R^4, … Each space R^n consists of a whole collection of vectors. R^5 contains all column vectors with five components. This is called “5-dimensional space”.
我們從最重要的向量空間開(kāi)始。它們由R ^ 1，R ^ 2，R ^ 3，R ^ 4，…表示。每個(gè)空間R ^ n由向量的整個(gè)集合組成。 R ^ 5包含具有五個(gè)分量的所有列向量。這被稱(chēng)為“ 5維空間”。

DEFINITION The space Rn consists of all column vectors v with n components.
定義:空間Rn由具有n個(gè)分量的所有列向量v組成。

The components of v are real numbers, which is the reason for the letter R. A vector whose n components are complex numbers lies in the space Cn .
v的分量是實(shí)數(shù)，這是字母R的原因。一個(gè)n個(gè)分量為復(fù)數(shù)的向量位于空間Cn中。

The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors-the whole plane. Each vector gives the x and y coordinates of a point in the plane: v = ( x, y).
向量空間R2由xy平面表示。 R2中的每個(gè)向量v具有兩個(gè)分量。 “空間”一詞要求我們考慮所有這些向量-即整個(gè)平面。每個(gè)向量給出平面中一個(gè)點(diǎn)的x和y坐標(biāo)：v =（x，y）。

Similarly the vectors in R 3 correspond to points ( x, y, z) in three-dimensional space. The one-dimensional space R 1 is a line (like the x axis). As before, we print vectors as a column between brackets, or along a line using commas and parentheses:
類(lèi)似地，R3中的向量對(duì)應(yīng)于三維空間中的點(diǎn)（x，y，z）。一維空間R 1是一條線(xiàn)（類(lèi)似于x軸）。和以前一樣，我們將向量打印為方括號(hào)之間的一列，或使用逗號(hào)和括號(hào)沿一行：
【P124有圖】

The great thing about linear algebra is that it deals easily with five-dimensional space. We don’t draw the vectors, we just need the five numbers (or n numbers).
線(xiàn)性代數(shù)的偉大之處在于它可以輕松處理五維空間。我們不繪制向量，我們只需要五個(gè)數(shù)字（或n個(gè)數(shù)字）。

To multiply v by 7, multiply every component by 7. Here 7 is a “scalar”. To add vectors in R 5, add them a component at a time. The two essential vector operations go on inside the vector space, and they produce linear combinations:
將v乘以7，將每個(gè)分量乘以7。這里7是“標(biāo)量”。要在R 5中添加向量，請(qǐng)一次添加一個(gè)分量。向量空間內(nèi)有兩個(gè)基本向量運(yùn)算，它們產(chǎn)生線(xiàn)性組合：

We can add any vectors in Rn, and we can multiply any vector v by any scalar c.
我們可以在Rn中添加任何向量，并且可以將任何向量v與任何標(biāo)量c相乘。

“Inside the vector space” means that the result stays in the space. If v is the vector in R 4 with components 1, 0, 0, 1, then 2v is the vector in R4 with components 2, 0, 0, 2. (In this case 2 is the scalar.) A whole series of properties can be verified in Rn. The commutative law is v + w = w + v; the distributive law is c( v + w) = cv + cw. There is a unique “zero vector” satisfying O + v = v. Those are three of the eight conditions listed at the start of the problem set.
“向向量空間中添加向量”表示添加后依然存在空間中。如果v向量是R 4中具有分量1、0、0、1的向量，則2v是R4中具有分量2、0、0、2的向量（在這種情況下2是標(biāo)量。）可以在Rn中驗(yàn)證。交換律是v + w =?? w + v;分配律是c（v + w）= cv + cw。存在一個(gè)滿(mǎn)足O + v = v的唯一“零向量”。這是問(wèn)題集開(kāi)始處列出的八個(gè)條件中的三個(gè)。

These eight conditions are required of every vector space. There are vectors other than column vectors, and there are vector spaces other than Rn, and all vector spaces have to obey the eight reasonable rules.
每個(gè)向量空間都需要這八個(gè)條件。除列向量外，還有其他向量；而除Rn以外，還有其他向量空間，所有向量空間都必須遵守8個(gè)合理規(guī)則。
A real vector space is a set of “vectors” together with rules for vector addition and for multiplication by real numbers. The addition and the multiplication must produce vectors that are in the space. And the eight conditions must be satisfied (which is usually no problem). Here are three vector spaces other than Rn:
實(shí)向量空間是一組“向量”，并且滿(mǎn)足向量加法和與實(shí)數(shù)相乘的規(guī)則。加乘之后的向量依然包含于該向量空，。并且必須滿(mǎn)足八個(gè)條件（通常沒(méi)有問(wèn)題）。這是Rn以外的三個(gè)向量空間：

M The vector space of all real 2 by 2 matrices.
F The vector space of all real functions f ( x).
Z The vector space that consists only of a zero vector.
M所有實(shí)2x2矩陣的向量空間。
F所有實(shí)函數(shù)f（x）的向量空間。
Z僅由零向量組成的向量空間。

In M the “vectors” are really matrices. In F the vectors are functions. In Z the only addition is O + 0 = 0. In each case we can add: matrices to matrices, functions to functions, zero vector to zero vector. We can multiply a matrix by 4 or a function by 4 or the zero vector by 4. The result is still in M or F or Z. The eight conditions are all easily checked.
在M中，“向量”實(shí)際上是矩陣。在F中，向量是函數(shù)。在Z中，唯一的加法是O + 0 =0。在每種情況下，我們可以做矩陣加法，做函數(shù)加法，做零向量加法。我們可以將矩陣乘以4或?qū)⒑瘮?shù)乘以4或?qū)⒘阆蛄砍艘?。結(jié)果仍為M或F或Z。八個(gè)條件都可以輕松核查。

The function space F is infinite-dimensional. A smaller function space is P, or P n, containing all polynomials a0 + a1 x + · · · + anxn of degree n.
函數(shù)空間F是無(wú)限維的。較小的函數(shù)空間為P或P n，其中包含度為n的所有多項(xiàng)式a0 + a1 x + … + + an^n。

The space Z is zero-dimensional (by any reasonable definition of dimension). Z is the smallest possible vector space. We hesitate to call it RO , which means no components- you might think there was no vector. The vector space Z contains exactly one vector (zero). No space can do without that zero vector. Each space has its own zero vector-the zero matrix, the zero function, the vector (0, 0, 0) in R3.
空間Z是零維的（這個(gè)定義是合理的）。 Z是最小的向量空間。我們猶豫將其稱(chēng)為RO，這意味著沒(méi)有分量-您可能會(huì)認(rèn)為沒(méi)有向量。向量空間Z恰好包含一個(gè)向量（零）。每個(gè)空間都有零向量。每個(gè)空間都有自己的零向量-零矩陣，零函數(shù)，R3中的向量（0，0，0）。
【書(shū)P125有圖】
Figure 3.1: “Four-dimensional” matrix space M. The “zero-dimensional” space Z.
圖3.1：“四維”矩陣空間M。“零維”空間Z。

Subspaces

At different times, we will ask you to think of matrices and functions as vectors. But at all times, the vectors that we need most are ordinary column vectors. They are vectors with n components-but maybe not all of the vectors with n components. There are important vector spaces inside Rn . Those are subspaces of Rn .
有時(shí)候，我們會(huì)要求您將矩陣和函數(shù)視為向量。但是大部分時(shí)間，向量指的是一般的列向量。它們是具有n個(gè)分量的向量-但可能不是所有具有n個(gè)分量的向量。 Rn內(nèi)部有重要的向量空間。這些是Rn的子空間。

Start with the usual three-dimensional space R 3. Choose a plane through the origin ( 0, 0, 0). That plane is a vector space in its own right. If we add two vectors in the plane, their sum is in the plane. If we multiply an in-plane vector by 2 or -5, it is still in the plane. A plane in three-dimensional space is not R2 (even if it looks like R2). The vectors have three components and they belong to R 3. The plane is a vector space inside R 3.
從通常的三維空間R 3開(kāi)始。選擇一個(gè)通過(guò)原點(diǎn)（0，0，0）的平面。該平面本身就是矢量空間。如果兩個(gè)在平面中的向量相加，那和也在平面中。平面內(nèi)一個(gè)向量乘以2或-5，則它仍在平面內(nèi)。三維空間中的平面不是R2（即使看起來(lái)像R2）。向量具有三個(gè)分量，它們屬于R 3。平面是R 3內(nèi)的向量空間。

This illustrates one of the most fundamental ideas in linear algebra. The plane going through (0, 0, 0) is a subspace of the full vector space R 3.
這說(shuō)明了線(xiàn)性代數(shù)中最基本的思想之一。經(jīng)過(guò)（0，0，0）的平面是整個(gè)向量空間R 3的子空間。

DEFINITION A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: If v and ware vectors in the subspace and c is any scalar, then
(i) v + w is in the subspace
(ii) cv is in the subspace.
定義向量空間的子空間是滿(mǎn)足兩個(gè)要求的向量集合（包括0）：如果子空間中的v和ware向量，而c是任意標(biāo)量，則
（i）v + w在子空間中
（ii）cv在子空間中。
定義：一個(gè)向量空間的子空間是滿(mǎn)足兩個(gè)條件的向量的集合。條件，如果v和w是子控件中的向量，c是任意標(biāo)量，那么（i）v + w在子空間中（ii）cv在子空間中。

In other words, the set of vectors is “closed” under addition v + w and multiplication cv (and dw). Those operations leave us in the subspace. We can also subtract, because -w is in the subspace and its sum with v is v - w. In short, all linear combinations stay in the subspace.
換句話(huà)說(shuō)，在以上兩種運(yùn)算中，結(jié)果依然在向量的集合中。減法依然如此，因?yàn)?w在子空間中，并且它與v的和是v-w。簡(jiǎn)而言之，所有線(xiàn)性組合的結(jié)果依然在子空間中。

All these operations follow the rules of the host space, so the eight required conditions are automatic. We just have to check the linear combinations requirement for a subspace.
所有這些操作都遵循主機(jī)空間的規(guī)則，因此八個(gè)必需條件是自動(dòng)的。我們只需要檢查子空間的線(xiàn)性組合要求即可。

First fact: Every subspace contains the zero vector. The plane in R^3 has to go through (0, 0, 0). We mention this separately, for extra emphasis, but it follows directly from rule (ii). Choose c = 0, and the rule requires Ov to be in the subspace.
Planes that don’t contain the origin fail those tests. Those planes are not subspaces.

Lines through the origin are also subspaces. When we multiply by 5, or add two vectors on the line, we stay on the line. But the line must go through(0,0,0).
Another subspace is all of R^3. The whole space is a subspace(of itself). Here is a list of all the possible subspaces of R3:
(L)Any line through(0,0,0)
(R^3)The whole space
§Any plane through(0,0,0)
(Z)The single vector(0,0,0)

If we try to keep only part of a plane or line, the requirements for a subspace don’t hold. Look at these examples in R2-they are not subspaces.
Example 1 Keep only the vectors(x,y) whose components are positive or zero(this is a quarter-plane). The vector(2,3) is included but(-2,-3) is not. So rule(ii) is violated when we try to multiply byc=-1. The quarter-plane is not a subspace.
Example 2 Include also the vectors whose components are both negative. Now we have two quarter-planes. Requirement(ii) is satisfied; we can multiply by any c. But rule(i) now fails. The sum of v=(2,3) and w=(-3,-2) is(-1,1), which is outside the quarter-planes. Two quarter-planes don’t make a subspace.
Rules(i) and(ii) involve vector addition v +w and multiplication by scalars c and d.The rules can be combined into a single requirement-the rule for subspaces:

A subspace containing v and w must contain all linear combinations cv+dw.

Example 3 Inside the vector space M of all 2 by 2 matrices, here are two subspaces:
(U) All upper triangular matices [ab|0d]
(D) All diagonal matrices [a0|0d].
Add any two matrices in U, and the sum is in U. Add diagonal matrices, and the sum is diagonal. In this case D is also a subspace of U! Of course the zero matrix is in these subspaces, when a,b, and d all equal zero.Z is always a subspace.
Multiples of the identity matrix also form a subspace.2I +3I is in this subspace, and so is 3 times 4I. The matrices cI form a"line of matrices"inside M and U and D.
Is the matrix I a subspace by itself? Certainly not.Only the zero matrix is.Your mind will invent more subspaces of 2 by 2 matrices-write them down for Problem 5.

3.2 The Nullspace of A: Solving Ax=0 and Rx=0

3.3 The Complete Solution to Ax=b

3.4 Independence, Basis and Dimension

3.5 Dimensions of the Four Subspaces

總結(jié)

以上是生活随笔為你收集整理的【人工翻译线代教材】Introduction to Linear Algebra BY Gilbert Strang【MIT】【线代】【翻译】的全部?jī)?nèi)容，希望文章能夠幫你解決所遇到的問(wèn)題。

如果覺(jué)得生活随笔網(wǎng)站內(nèi)容還不錯(cuò)，歡迎將生活随笔推薦給好友。