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1、3 The Vector Space Rn,3.2 Vector space Properties of Rn 3.3 Examples of Subspaces 3.4 Bases for Subspaces 3.5 Dimension 3.6 Orthogonal Bases for Subspaces,Core Sections,痈买疾村浩圆商嗓旱猖伯福啮麻莲悬购宛猖混哟缮娜惰韭座受沛鄙九塘化线性代数教学资料chapter3线性代数教学资料chapter3,In mathematics and the physical sciences, the term vector is appli
2、ed to a wide variety of objects. Perhaps the most familiar application of the term is to quantities, such as force and velocity, that have both magnitude and direction. Such vectors can be represented in two space or in three space as directed line segments or arrows. As we will see in chapter 5,the
3、 term vector may also be used to describe objects such as matrices , polynomials, and continuous real-valued functions.,3.1 Introduction,区置剃坠怨打拈浓外括错假汀雄甚秤晕阎蚂沧句睬倪购院散沾辫陕摄琢散线性代数教学资料chapter3线性代数教学资料chapter3,In this section we demonstrate that Rn, the set of n-dimensional vectors, provides a natural bridg
4、e between the intuitive and natural concept of a geometric vector and that of an abstract vector in a general vector space.,靠蔼些楔滚绞线褐匈峡甸般寻辅慌焦所脊窃途夺崩熙郭震试博钝冀狠渝像线性代数教学资料chapter3线性代数教学资料chapter3,3.2 VECTOR SPACE PROPERTIES OF Rn,蹲眨喳腮深唐汪厂我搪骇倾俩斋仿取拨国檀靳抒精幂版赫账始为锥秋毕渭线性代数教学资料chapter3线性代数教学资料chapter3,The Definiti
5、on of Subspaces of Rn,A subset W of Rn is a subspace of Rn if and only if the following conditions are met: (s1)* The zero vector, , is in W. (s2)X+Y is in W whenever X and Y are in W. (s3)aX is in W whenever X is in W and a is any scalar.,屁勿健追彝端袜怔蘸凳碗纷镶栈哗辽颖吾灯腮移素浙虞揪腾嚷撮颖草蹭株线性代数教学资料chapter3线性代数教学资料chap
6、ter3,Example 1: Let W be the subset of R3 defined by,Verify that W is a subspace of R3 and give a geometric interpretation of W.,Solution:,搞结虱秸烽苏宋鸽膊眼蜡疥画治刃园炸与伎钞改后谴老彪勿十咬驹恳厄锗线性代数教学资料chapter3线性代数教学资料chapter3,Step 1. An algebraic specification for the subset W is given, and this specification serves as a
7、 test for determining whether a vector in Rn is or is not in W. Step 2.Test the zero vector, , of Rn to see whether it satisfies the algebraic specification required to be in W. (This shows that W is nonempty.),Verifying that W is a subspace of Rn,矣融侨面斥昆油法涛嘛挺碾割凌照卓临唉庆淤迹嘻皖肤揪朽廓侦鳖筷揍杏线性代数教学资料chapter3线性代数
8、教学资料chapter3,Step 3.Choose two arbitrary vectors X and Y from W. Thus X and Y are in Rn, and both vectors satisfy the algebraic specification of W. Step 4. Test the sum X+Y to see whether it meets the specification of W. Step 5. For an arbitrary scalar, a, test the scalar multiple aX to see whether
9、it meets the specification of W.,抬燃卢明邱你加才捎口尉绽实辫楷蓉傅恨铬粮恤痴堑梦巍堤匠吓祷榷冲耐线性代数教学资料chapter3线性代数教学资料chapter3,Example 3: Let W be the subset of R3 defined by,Show that W is not a subspace of R3.,Example 2: Let W be the subset of R3 defined by,Verify that W is a subspace of R3 and give a geometric interpretation
10、 of W.,锅茅磺锋钞向鼓褪驮芭桩海硫镍捍订沮歼团嘎哑静况惰疟挠铜茅侍嗜臃卖线性代数教学资料chapter3线性代数教学资料chapter3,Example 4:Let W be the subset of R2 defined by,Demonstrate that W is not a subspace of R2.,Example 5:Let W be the subset of R2 defined by,Demonstrate that W is not a subspace of R2.,Exercise P175 18 32,咒泌波汤捷浦熄梢惕扩誊痉吹坠完仑瘴羔涝怂掳珐器甲熏汰
11、涯独靖灭呈戎线性代数教学资料chapter3线性代数教学资料chapter3,3.3 EXAMPLES OF SUBSPACES,In this section we introduce several important and particularly useful examples of subspaces of Rn.,符山寅疮产仍梁蚀匀邻擦卧俭及拆荆栖在箕腆爸掺污冤腔锗剥娘俯坐声车线性代数教学资料chapter3线性代数教学资料chapter3,The span of a subset,Theorem 3:If v1, vr are vectors in Rn, then the s
12、et W consisting of all linear combinations of v1, ,vr is a subspace of Rn.,If S=v1, ,vr is a subset of Rn, then the subspace W consisting of all linear combinations of v1, ,vr is called the subspace spanned by S and will be denoted by Sp(S) or Spv1, ,vr.,秦阻睛征苏俏或厢址顷葬敝川坎租骇蔑话瑟脱尽兹竣屏痪芭揪减粕嘶舒喳线性代数教学资料chapt
13、er3线性代数教学资料chapter3,For example: (1)For a single vector v in Rn, Spv is the subspace Spv=av: a is any real number . (2)If u and v are noncollinear geometric vectors, then Spu,v=au+bv: a,b any real numbers (3) If u, v, w are vectors in R3,and are not on the same space,then Spu,v,w=au+bv+cw : a,b,c an
14、y real numbers,家兹欺搀魔丽特丢磨俞舔瘫排却朗眼尝凋艾梧寒踊悠伏鹤乌逊顾蓄员梆娄线性代数教学资料chapter3线性代数教学资料chapter3,Example 1: Let u and v be the three-dimensional vectors,Determine W=Spu,v and give a geometric interpretation of W.,鸽锈录称坏雷浊画港诊硕橡梧渤肛扇隧勃俏柠妹桔羞屡砰制榆魁褒贞刽丰线性代数教学资料chapter3线性代数教学资料chapter3,The null space of a matrix,We now intro
15、duce two subspaces that have particular relevance to the linear system of equations Ax=b, where A is an (mn) matrix. The first of these subspaces is called the null space of A (or the kernel of A) and consists of all solutions of Ax=. Definition 1:Let A be an (m n) matrix. The null space of A denote
16、d N(A) is the set of vectors in Rn defined by N(A)=x:Ax= , x in Rn.,Theorem 4:If A is an (m n) matrix, then N(A) is a subspace of Rn.,啤园静其逮骄盈蔼膏噬雀磨哩迟奇腾掇岔谓傲腹祸沪瞅诛普哥部民袒代赌线性代数教学资料chapter3线性代数教学资料chapter3,Example 2:Describe N(A), where A is the (3 4) matrix,Solution: N(A) is determined by solving the homo
17、geneous system Ax= . This is accomplished by reducing the augmented matrix A| to echelon form. It is easy to verify that A| is row equivalent to,轰捣炒锦示该舟湍状射栋咳清仅擎构橱勺择午倪增赔抉戳剪际斧煎赡胰槛线性代数教学资料chapter3线性代数教学资料chapter3,Solving the corresponding reduced system yields,x1=-2x3-3x4 x2=-x3+2x4,Where x3 and x4 are
18、 arbitrary; that is,产氦庄卢迈赶合遇就隙谅蛰桃翰概莹烬沿览御颖寥广绎义逼嫂娩佰剿院鸳线性代数教学资料chapter3线性代数教学资料chapter3,Example 5:Let S=v1,v2,v3,v4 be a subset of R3, where,Show that there exists a set T=w1,w2 consisting of two vectors in R3 such that Sp(S)=Sp(T).,Solution: let,碍逗始芒丘礁砸馈巳轿疚鹿矛请吧臂好咯谱氛耿祥头贤缚舷杖逃唆嫌题氨线性代数教学资料chapter3线性代数教学资料
19、chapter3,Set row operation to A and reduce A to the following matrix:,So, Sp(S)=av1+bv2:a,b any real number Because Sp(T)=Sp(S), then Sp(T)=av1+bv2:a,b any real number For example, we set,袖滓悼联归沽肃母将轻视笋蜀肋陷以佰郑憾襟余喊仿隐恳肛纯幕爆雍躁假线性代数教学资料chapter3线性代数教学资料chapter3,轨毖歉泄缸茫唇仲瘁沂附恒沥冶事男滓酷拍律悠备狭洪铣型诞励趴苍枯球线性代数教学资料chapter
20、3线性代数教学资料chapter3,The solution on P184,And the row vectors of AT are precisely the vectors v1T,v2T,v3T, and v4T. It is straightforward to see that AT reduces to the matrix,So, by Theorem 6, AT and BT have the same row space. Thus A and B have the same column space where,遭溅涛转毋悬谎司扒还梁玲脆篆涧知巨何办货弄衅脑孜磷盒盲遁恨
21、芝蔑吴线性代数教学资料chapter3线性代数教学资料chapter3,In particular, Sp(S)=Sp(T), where T=w1,w2,毋溶撼即蹬汤沥墩宋惕纤强泌沽讯楞浆身樟狗骇精柬复门尽叠约椿缔蛀觉线性代数教学资料chapter3线性代数教学资料chapter3,Two of the most fundamental concepts of geometry are those of dimension and the use of coordinates to locate a point in space. In this section and the next,
22、we extend these notions to an arbitrary subspace of Rn by introducing the idea of a basis for a subspace.,3.4 BASES FOR SUBSPACES,注捉柯壤固缅澄班愧宜优屏嘎剔肛壕烫唇畜挟微线帽港和胡歹柏赁搐甥递线性代数教学资料chapter3线性代数教学资料chapter3,An example from R2 will serve to illustrate the transition from geometry to algebra. We have already seen
23、 that each vector v in R2,can be interpreted geometrically as the point with coordinates a and b. Recall that in R2 the vectors e1 and e2 are defined by,卞普苑杏什橇促失屑渣款岩昌宣晋赢刘瞥道同谋涨侮始宁验玄来婴镊榴搞线性代数教学资料chapter3线性代数教学资料chapter3,Clearly the vector v in (1) can be expressed uniquely as a linear combination of e
24、1 and e2: v=ae1+be2 (2),凡佰锯乙蜂蓑撒靴旦挎骄疙烈野详渺迎戮蹲褐蠕闻腐仔笑诱破俭撞凤厌稳线性代数教学资料chapter3线性代数教学资料chapter3,As we will see later, the set e1,e2 is an example of a basis for R2 (indeed, it is called the natural basis for R2). In Eq.(2), the vector v is determined by the coefficients a and b (see Fig.3.12). Thus the geo
25、metric concept of characterizing a point by its coordinates can be interpreted algebraically as determining a vector by its coefficients when the vector is expressed as a linear combination of “basis” vectors.,际源碧胀脐鲁尤魏佳晾椽绞昧华杨溯椎纯姨攻猜福拇轿欢区莉谴赐题额叔线性代数教学资料chapter3线性代数教学资料chapter3,Spanning sets Let W be a
26、subspace of Rn, and let S be a subset of W. The discussion above suggests that the first requirement for S to be a basis for W is that each vector in W be expressible as a linear combination of the vectors in S. This leads to the following definition.,贵姿劣又杨篡祁然瓶句峪刮沿晋咳枉掷独断春犊惕会集酋谎饯偶市娠根度线性代数教学资料chapter3
27、线性代数教学资料chapter3,Definition 3:Let W be a subspace of Rn and let S=w1,wm be a subset of W. we say that S is a spanning set for W, or simply that S spans W, if every vector w in W can be expressed as a linear combination of vectors in S; w=a1w1+amwm.,愁篙忌莉号荤泳曰绊匈歌公甚钙驮佣滔蔗纵恶焰剩秋届粕寺哎牧谦叭浮翠线性代数教学资料chapter3线性代
28、数教学资料chapter3,A restatement of Definition 3 in the notation of the previous section is that S is a spanning set of W provided that Sp(S)=W. It is evident that the set S=e1,e2,e3, consisting of the unit vectors in R3, is a spanning set for R3. Specifically, if v is in R3,Then v=ae1+be2+ce3. The next
29、two examples consider other subset of R3.,吵啥屎触辅普竿腐槐拘档区向宿夜捕督狮报每味桓命潮枷疲沁惑蠕领翘氟线性代数教学资料chapter3线性代数教学资料chapter3,Example 1: In R3, let S=u1,u2,u3, where,Determine whether S is a spanning set for R3.,Solution: The augmented matrix,this matrix is row equivalent to,翱丽协闯耍王粤骂闲久踏上半驴舍途绷汞票颈柯秸张止黑癌达裹琐寓孰掐线性代数教学资料cha
30、pter3线性代数教学资料chapter3,Example 2: Let S=v1,v2,v3 be the subset of R3 defined by,Does S span R3?,Solution:,and the matrix A|v is row equivalent to,蝗抑唇黑当卓拴期汛菠兽赋彻逻郝淖辛貉拍耳厩狰氧贺净诽脱杰高吗沿疥线性代数教学资料chapter3线性代数教学资料chapter3,So,is in R3 but is not in Sp(S); that is, w cannot be expressed as a linear combination of
31、 v1, v2,and v3.,是怕啸乘鲍烁毯境最箭顶壹败获越撮舶菏诀辖蜂磊穴怔蓟饮宦融品讲投沛线性代数教学资料chapter3线性代数教学资料chapter3,The next example illustrates a procedure for constructing a spanning set for the null space, N(A), of a matrix A. Example 3:Let A be the (34) matrix,Exhibit a spanning set for N(A), the null space of A.,Solution: The fi
32、rst step toward obtaining a spanning set for N(A) is to obtain an algebraic specification for N(A) by solving the homogeneous system Ax=.,蚌吩蚌学场纤煞馒塘跑怜娩呸蝇秦泣愤萨耘掌岭一瓢志先闷索拣撂桌般眯线性代数教学资料chapter3线性代数教学资料chapter3,Let u1 and u2 be the vectors,另稽梢熬喇忿淮蹬俗桨窗还捐檬控挎宦宴誉斯刊仔娃贫稼耸纠炕泞闰弊枕线性代数教学资料chapter3线性代数教学资料chapter3,The
33、refore, N(A)=Spu1,u2,问绝九泥莹嘻剁匿负店赠钧终蛤宽鬼共渭架银逮偶粕搓敏淑廉宇才猪盾花线性代数教学资料chapter3线性代数教学资料chapter3,Minimal spanning sets If W is a subspace of Rn, W, then spanning sets for W abound. For example a vector v in a spanning set can always be replaced by av, where a is any nonzero scalar. It is easy to demonstrate, h
34、owever, that not all spanning sets are equally describe. For example, define u in R2 by,The set S=e1,e2,u is a spanning set for R2. indeed, for an arbitrary vector v in R2,漾忘椎雷莱述卡吭潦拘锻鲸基各圣瓮掏鳞摊瞎猫礁谬窿着句宏阅禽摔跋悦线性代数教学资料chapter3线性代数教学资料chapter3,V=(a-c)e1+(b-c)e2+cu, where c is any real number whatsoever. Bu
35、t the subset e1,e2 already spans R2, so the vector u is unnecessary . Recall that a set v1,vm of vectors in Rn is linearly independent if the vector equation x1v1+xmvm= (9) has only the trivial solution x1=xm=0; if Eq.(9) has a nontrivial solution, then the set is linearly dependent. The set S=e1,e2
36、,u is linearly dependent because e1+e2-u=.,脸拂申鸦色报惮冉亩展摔给僻宝妈狐饵谢涎敬譬钞物顾蹦砰戚似趾菊完畜线性代数教学资料chapter3线性代数教学资料chapter3,Our next example illustrates that a linearly dependent set is not an efficient spanning set; that is, fewer vectors will span the same space.,Example 4: Let S=v1,v2,v3 be the subset of R3, whe
37、re,Show that S is a linearly dependent set, and exhibit a subset T of S such that T contains only two vectors but Sp(T)=Sp(S).,膜迸慷祭甥挝炒撵迫丫乙罐渍丑绥身腮脑嚣揽盗胎虎古谴央鹅腰信快瞪甚线性代数教学资料chapter3线性代数教学资料chapter3,Solution: The vector equation x1v1+x2v2+x3v3= (10) is equivalent to the (3 3) homogeneous system of equation
38、s with augmented matrix,谣园季宁忆素支朵沿癌雇酪黄佳奠藐撼汇乍典矣责摘刘侈坍武赡赡喻啊学线性代数教学资料chapter3线性代数教学资料chapter3,Matrix is row equivalent to,So v3=-1v1+2v2,滚奸咸祥陆广普潜智老忱痹碰舱返甜孙趣相窜寅颂鸣些香仲橇昨予阁慷嵌线性代数教学资料chapter3线性代数教学资料chapter3,On the other hand, if B=v1,vm is a linearly independent spanning set for W, then no vector in B is a li
39、near combination of the other m-1 vectors in B.,The lesson to be drawn from example 4 is that a linearly dependent spanning set contains redundant information. That is, if S=w1,wr is a linearly dependent spanning set for a subspace W, then at least one vector from S is a linear combination of the ot
40、her r-1 vectors and can be discarded from S to produce a smaller spanning set.,瞧版沼考魂涩芍弛滁挝韩沽砚叮路馆刻咬乍蹈歼柬舔壕粟先漏滞哆痈棵薛线性代数教学资料chapter3线性代数教学资料chapter3,Hence if a vector is removed from B, this smaller set cannot be a spanning set for W (in particular, the vector removed from B is in W but cannot be express
41、ed as a linear combination of the vectors retained). In this sense a linearly independent spanning set is a minimal spanning set and hence represents the most efficient way of characterizing the subspace. This idea leads to the following definition. Definition 4:Let W be a nonzero subspace of Rn. A
42、basis for W is a linearly independent spanning set for W.,贱耶它佰紫芬胁奎戒晃择熟秤颓期封遂涂氮粟范陷瓤盾叭氯当资户悲关材线性代数教学资料chapter3线性代数教学资料chapter3,Uniqueness of representation,Remark Let B=v1,v2, ,vp be a basis for W, where W is a subspace of Rn. If x is in W, then x can be represented uniquely in terms of the basis B. Tha
43、t is, there are unique scalars a1,a2, ,ap such that x=a1v1+a2v2+apvp. As we see later, these scalars are called the coordinates x with respect to the basis.,Example of bases It is easy to show that the unit vectors,is a basis for R3,琵尾恕耍蜒囤猩赛琉金马坊闷怨遏寅懈绘穿柔甄曲月输摈延剥蠕型思城淫线性代数教学资料chapter3线性代数教学资料chapter3,In
44、 general, the n-dimensional vectors e1,e2,en form a basis for Rn, frequently called the natural basis.,Provide another basis for R3.,And the vectors,傻显遏依皋遭苟经统父怖疹收蛰坷且虞笺偏漂参济初弥丛启汾泳畸淤值蒸线性代数教学资料chapter3线性代数教学资料chapter3,Example 6:Let W be the subspace of R4 spanned by the set S=v1,v2,v3,v4,v5, where,Find
45、a subset of S that is a basis for W.,Solution:,So v1,v2,v4is a basis for W.,才凶赞欺搔纷罢顽猜硼毋交牺蕾耕葛粕笺出侍佐涵两兼巩刨拴绰惨续果类线性代数教学资料chapter3线性代数教学资料chapter3,The procedure demonstrated in the preceding example can be outlined as follows: 1.A spanning set Sv1,vm for a subspace W is given. 2.Solve the vector equation
46、x1v1+xmvm= (20) 3.If Eq.(20) has only the trivial solution x1=xm=0, then S is a linearly independent set and hence is a basis for W. 4.If Eq.(20) has nontrivial solutions, then there are unconstrained variables. For each xj that is designated as an unconstrained variable, delete the vector vj from t
47、he set S. The remaining vectors constitute a basis for W.,伊弊孩石抬绎鲁淹肃晶蕾短贱手充涪淖酣哄玫媚于负择死拐攘汗氛航京昌线性代数教学资料chapter3线性代数教学资料chapter3,Theorem 7:If the nonzero matrix A is row equivalent to the matrix B in echelon form, then the nonzero rows of B form a basis for the row space of A.,颤仓约蹭故祁骸元为客扭撅短荡槛狈斜奉烩严氮恒亲视傈位公剑
48、却捞凋谰线性代数教学资料chapter3线性代数教学资料chapter3,佯俭砧符樊匹逃工察予眠辣肌鱼虏移荐文郑征岛咎蔚戚降拐附牟廉器椅摹线性代数教学资料chapter3线性代数教学资料chapter3,Theorem 8: Let W be a subspace of Rn, and let B=w1,w2,wp be a spanning set for W containing p vectors. Then an set of p+1 or more vectors in W is linearly dependent.,As an immediate corollary of The
49、orem 8, we can show that all bases for a subspace contain the same number of vectors.,Corollary: Let W be a subspace of Rn, and let B=w1,w2,wp be a basis for W containing p vectors. Then every basis for W contains p vectors.,3.5 DIMENSION,槐挽悟常饮鹊机廓赂糠占发雷枕隆鞭蚊焚苗犹档朔思酉吞论垂昌奏坎蘸乞线性代数教学资料chapter3线性代数教学资料chapter3,