線性代數(shù)及其應(yīng)用術(shù)語(yǔ)要點(diǎn)中英對(duì)照

1、Chapter 1-Review1. 線性方程組 Systems of Linear Equations (Linear System)P3 】關(guān)鍵詞：coefficient 系數(shù)P2; constant term 常數(shù)(項(xiàng))講5C-P1； linear equation 線性方程P2； variable 未知數(shù)(或變?cè)?有個(gè)方程個(gè)未知數(shù)仗訃)的姣性方程組可表示為：1) + a2 + + 知片=(/i.a b 為 m 維列向)3) Ax=b (A是mx 矩陣；x tb為m維列向)4) Augmented “tairix(増 廣矩陣)-(其中第/a j勻列是變?cè)鸬南禂?shù))2線性方程組解的憎況(S

2、olution Status)【P41)2)No solution無(wú)解Has Solution 有解a) Exactly one solution (unu/ne sohitioit)b) Infinite many sobitions無(wú)窮多解唯一解3階梯形(Echelon Forms)P14關(guān)鍵leading entry 先導(dǎo)元素P14; pivot position 主元位置P16;1) 3 conditions of echelon form matrix階梯形矩陣的三個(gè)條件(缺一不可)：a) A zero nnv is not above on any nonzero nnr所有非零行

3、都在菲行上部b) Each leading entry of a rmr is on the right of the leading entry of the previous zwr 毎彳亍的先 導(dǎo)元素都在上一行先導(dǎo)元素的右邊c) In each colunui, an entry below the leading entry is 0與先導(dǎo)元素同列且在其下部的元素全 為02) 2 additional conditions of Reduced Echelon Forms 簡(jiǎn)化階梯形的額外兩個(gè)性質(zhì)：a) Tlie leading entry of each nonzero zw is

4、1每一非零行的先導(dǎo)元素都是 1b) Each leading 1 is the ONLY nonzero entry of its column先導(dǎo)元素是其所在列唯一非零元素注：與線性方程組結(jié)合： f . f No free variables unique solution free variable * infinitely many solutionsfree variable至少有一個(gè)自由變注：結(jié)合簡(jiǎn)化階梯形采用反證法輕松搞定！Additionally,此外：if r = #pivot positions, p = free variables, n =體variables then

5、r+P = n,#0 - number of Q (J 的個(gè)數(shù))6. 非齊次線性方程組解的結(jié)構(gòu)定理(Structure of Solution Set ofP53Nonhomogeneous System)關(guān)鍵詞：nonhomogeneous system非齊次線性方程組P50；Let Vo be a solution of a nonhomogeneous system Ax = b.Let H be the set of general solutions of the corresp on ding homoge neous system Ax 二 0. Suppose the solu

6、tion set of Ax = b is SThen S = H+ Vo如果是非齊次線性方程組Ax = b的一個(gè)解，H是對(duì)應(yīng)齊次線性方程組M = 0的通解。(XU二 0也稱為Ax = b的導(dǎo)出組)則Ax = b的通解是voProofApparently, vhw H, (h+ vO) e S;so, HcS; (1)Now, bv e S, v- Vo e H9 since A (v v。)= Av Av。= b-b = 0;Because v vo + vH +VoConsequently: v e Voand thus ScH (2)Given (1) and (2), we now h

7、ave S = H.E.g.: (Examples 5.1 and 5.2)Ax-0:f.一 _.r *jX)丄上4T*厶 U| ZTi +QP iXj H；x-. Kp 為標(biāo))有解三 b 是 aba.an 的線性組合8. 線性無(wú)關(guān)/ 相關(guān)(Linear Independent / Dependent)P65關(guān)鍵詞trivial solutions非零解/非平凡解P51；莎m維空間P281) Definition P65Vector set qj, “2，心 “ linear dependent ifxjaj + Vjflj + + “曲并二 0 has only the trivial so

8、lution (XjX2 .xn are all 0)如果方程組+%劉+.+*”= 0只有零解(.v2.v2 .x全是0),則心， a2,.anttt 性無(wú)關(guān)。Vector set aj, 2, “幵石 linear independent tfxjaj + + xn= 0 if .xn are not all 0.若方程組xiaj+x2fl2+有非零解(xjx2 .X.,不全是O),則向量組“ a2,.an線性相關(guān)。2) Theorem 7 Characterization of Linearly Dependent定理7線性相關(guān)和線性組合的關(guān)系定理 P68Vector set qi, a2

9、,.an) is linear dependent Exist vector a2(l i h), which is a linear conibiiialioii of the other vectors向組(alfa2f.an線性相關(guān)存在某向量心(區(qū))是其它向的線性組合注：由線性相關(guān)定義Xjaj +x申：+ . +%”= 0 , xjx2 . xn不全是0則線性相關(guān)。設(shè)X： H 0(7 0 把移到等式另一邊yfli = - ( Xfi +七a： + . + xffln ),然后兩邊除以Xj (因?yàn)殪鼿 0 )即得證向量al i m r (1 i /*), which is a linear

10、 combination of the other vectors如果向組中向個(gè)數(shù)大于向的維數(shù)皿，則向組線性相關(guān)。注：不知如何證明？看本表第5項(xiàng)100遍 o4) Theorem 9 Vector set qi, a2,.an is linear dependent if there exists 0 ( 1 i )a2f.an) ,a,= 0 ( 1 i alf a1.an線性相關(guān)汪：還是不知如何證明？看本格上面的定義100 JB o9. 等價(jià)定理(Theorem。P43關(guān)鍵詞：Rr,m 維空間P28; subset of Rrn spanned (or generated) by % j 由

11、 v.vp 張成(或生成的)的的子空間P35;1) For each b in 時(shí)；the system Ax = b has a solution對(duì)于 對(duì) 中的每一個(gè)向量b,線性方程組4丫 = b都有一個(gè)解2) Each b in P “ a linear combination of die cobinuis of A. 時(shí)中的毎一個(gè)向 b 都是矩陣 A 的 列向量的線性組合3) The coliinuis of A sp(ui - 矩陣 A 的列向生成4) The matrix A has a pivot position in every矩陣A毎一行都有一個(gè)主元位注：。3)根據(jù)定義顯然

12、成立；4)可用定理2采用反證法10. 補(bǔ)充齊次方程組基礎(chǔ)解系定理(Additional Theorem of basic solutions of aP43homogenous linear system)Proof:Suppose v； v5 . vpare the basic solutions of a homogeneous linear system Ax = 0 Then, we know that there are p Jtqq variables Ax-0 (為什么,看本衣第S項(xiàng))Let ciV + c2 + +c”Yp= v / where c厶 . cn are scal

13、ars.We knoxv that m each vector vt(l i p), there is a 1 conesponding to the position of the i-th free vanable In addition, each element in that portion in the other vector is 01Position of the ith free variable00MVConsequently, the element in this position of the vector vis ct Therefore, for vector

14、v to be a 0 vector, cj, c2, cnmust all be 0. uChapter 2matrix algebraP105矩陣代數(shù)matrix operationsPIO 刀矩陣的運(yùn)算main diagonal of matrixPIO 刀矩陣的主對(duì)角線diagonal matrixPIO 刀對(duì)角矩陣identity matrix lnP45+ PIO刀n x n單位矩陣matrix additionPIO 刀矩陣加法scalar multiplicationP109數(shù)乘（矩陣）matrix multiplicationP109矩陣乘法If A is an m x n

15、matrix, and B is an n x p matrix with columns b 仙, then the product of AB is the m x p matrix whose columns are Abi.AbpP110A: m x n矩陣B: n x p矩陣，矩陣的各列向量為bbphAB = AbjAb2 AbpThe vector in column j of AB is a linear combination of all the column vectors ai. an of A (weights are the entries of the corres

16、ponding byColumn of B)P110矩陣AB的第j列Vj都是A的所有列 向量血的線性組合.（其中各個(gè) 權(quán)是B中對(duì)應(yīng)列匕的元素）Theorem Rules for Matrix OperationA: m x n matrixB, C： matrices whose sizes in each row of the following allow the addition and multiplication in that row kz t: scalarP108+ P113矩陣運(yùn)算規(guī)則A: m x n矩陣B, C:在每行中，尺寸都符合那行加 法和乘法定義的矩陣k, t：標(biāo)量1

17、) Addition and scalar multiplication A+ B = B +A(A + B) + C = A + (B + C)A + 0 = Ak(A + B) = kA+kB (k+t| A = kA + tA k(tA) = (kt) A1）矩陣加法和數(shù)乘2) Multiplication A|BC| = (AB)C A(B+C) = AB +AC (B+CJA = BA +CA k(AB|-(kA)B = A(kB) UA = A = Alm2）矩陣乘法commuteP113可交換（矩陣乘法）Warnings:In general AB H BA AB=AC 壬 B

18、= CAB=O 豐 A - 0 or B - 0P114transpose of a matrixPH5矩陣的轉(zhuǎn)置Theorem 3 TranspositionA: m x n matrixAt: transpose of matrix AB: matrix whose size in each row of the following allow the addition and multiplication in that row k: scalar(AT)T = A|A + B) t = At + Bt |kA)T = kAT (AB)t = BtAtinvertibleP119（矩陣）

19、可逆的matrix inverseP119矩陣的逆singular matrixP119奇異矩陣nonsingular matrixP119非奇異矩陣Theorem 4 necessary and sufficient condition for a 2 x 2 matrix is invertibleLet A = adbc 工 0, then A is invertible3ndA=”“Lc alTheorem 4, A is invertible Iff det AHO(where det A = ad-bc)P119二階方陣人=:3 可逆的充要條件 ad-bc H 0或記作|A| H

20、0Theorem 5If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the unique solution x = A-1bP120定理5系數(shù)為n階可逆方陣A的線 性方程組Ax=b的解的情況定理若A是一個(gè)n階可逆矩陣，那么對(duì)于 n維空間R”中的每一個(gè)列向量b方 程組Ax = b都有唯一解x = A 2bTheorem 6 Rules ofA, B: n x n invertible matricesP121定理6矩陣的逆運(yùn)算規(guī)則(A1)1 = A (AB)1 = BW IaT

21、= (A1)7elementary matrixP122初等矩陣If an elementary row operation is performed on matrix A, the resulting matrix can be written as EA, where the m x m matrix E is created by performing the same row operation on ImP123左乘初等矩陣等價(jià)于進(jìn)行一次與初等矩陣一樣的行初等 變換Proof idea:Prove that each of the 3 kinds of row operations

22、, ifoil11 11191 f 1performed on a matrix A ml a*Q 土 IS thsame as left multiply the elementary matrix.three corresponding取：A乞o些=匂A, where Ed = 1巧o巧Theorem 7.An nxn matrix A is invertible iff A is row equivalent to and in this case, any sequence of elementary row operations that reduces A to 人 also tr

23、ansform into A 1P123定理7可逆矩陣判斷定理一個(gè)mm矩陣A是可逆的當(dāng)且僅當(dāng) A行等價(jià)于In（就是說(shuō)A可以行化 簡(jiǎn)成人）。并且，在這種情況下，任 何一系列把A行化簡(jiǎn)成厶的操作， 都可以把人轉(zhuǎn)化成A*1Algorithm for finding A Row reduce the augmented matrix A | I, if A is row equivalent to 1, then A | I is row equivalent to I | A I Otherwise, A is not ivertible.P124用初等行變換求逆矩陣：把增廣矩陣A | I化簡(jiǎn)，如果

24、A行等 價(jià)于單位陣,則A | I能化簡(jiǎn)成I | A1,否則A不可逆.Theorem 8. Invertible matrix theoremThe following statements are equivalent.a. A is an invertible matrix.b. A is row equivalent to the nxn identity matrix.c. A has n pivot positions.d. The equation Ax = 0 has only the trivialP129可逆矩陣性質(zhì)定理I、列斷言等價(jià)a. A是可逆的b. A行等價(jià)J：一個(gè)n階單

25、位陣。c. A有n個(gè)主尤位置。d. 矩陣方程Ax = 0僅有平凡解（零f. The linear transformation x i Ax isf. 線性變換XI Ax是一對(duì)一的。one-to-one.g. The equation Ax= b has only one solution forg-對(duì)珅中任意的一個(gè)向星b,矩陣方each b in Rn.程Ax= b有唯一解。h. The columns of A span Rn.h.A的列張成IT.i. The linear transformation x i Ax mapsI.線性變換X Ax把R”映射到RSRn to Rn.J-存在一

26、個(gè)n x n矩陣C使CA =1.j. There is an n x n matrix C such that CA = I.k.存在一個(gè)n x n矩陣D使AD =1.k There is an n x n matrix D such that AD -1.1. Ar is an invertible matrix.1.At是可逆的.partitioned matrix (block matrix)P134分塊矩陣multiplication of partitioned matricesP135分塊矩陣的乘法Partitions of A and B should be conformabl

27、e forP136A和B的分塊矩陣要相乘的話，Ablock multiplication和B的分法應(yīng)遵從矩陣乘法定義The column partition of A matches the rowA的列分法應(yīng)與B的行分法一致partition of B（左邊大小列=右邊大小行）Theorem 10 column-row expansion of ABP137定理10 AB乘法的列行展開e.e.If A is an m x n matrix and B is an n x p matrix thensolution.The columns of A form a linearly indepe

28、ndent set.解A的列形成一個(gè)線性無(wú)關(guān)集。AB = coli(A) col2(A) . coln(A)row2(F) rown(BcoljfA) roWtfB) +.+ coln(A) rown(B)subspaceP168子空間column space of AColA = all linear combinations of the columns of A =kjBi + . + knan (ki(1in)6R)P169A的列牢間ColA = A的所有列的線性組合形成的 向最的集合null space of ANul A = all solutions to the homogen

29、eous equationAx =0P169A的零空間Nul A =齊次線性方程組Ax = 0的 通解Theorem 12. Theorem for null space of AThe null space of an m x n matrix A is a subspace of R”P170A的零空間定理mxn矩陣A的零空間是R”的子空間 （這是因?yàn)锳x = 0的解向最是n維 的，所以它是n維空間的了空間）Equivalently, the set of all solutions to a system Ax= 0 of m homogeneous linear equations i

30、n n unknowns is a subspace of Rn.也就是說(shuō)，有著m個(gè)方程n個(gè)未知數(shù) 的方程組Ax=0的通解是R的子空間.basisP170基8/12P172 A的列空間定理A的主元列形成了 A的列空間的一個(gè)基。Theorem 13. Theorem for column space of AThe pivot columns of a matrix A form a basis for the column space of Acoordinate vector of x ( relative to B)dimension of a subspaceThe dimension

31、of a nonzero subspace H# denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero subspace is 0.rankTheorem 14. The Rank TheoremIf a matrix A has n columns then rank A dim NulA = nTheorem the invertible matrix theoremm.The columns of A form a basis of Rn.n.Col A= Rn.o

32、.dim Col A = n.p.rank A = n.q.NulA = 0r.dim Nul A = 0P176 X相對(duì)J * B的坐標(biāo)向量（對(duì)照解析幾何中，相對(duì)J：x軸,y軸,z 軸的坐標(biāo)）P177子空間的維數(shù)非零子空間的維數(shù)，用dimH衷示， 它是H的任意一個(gè)甚中，向靈的個(gè) 數(shù)。零子空間的維數(shù)定義成0（注意：與向量的維數(shù)區(qū)別?。㏄178秩定理14矩陣的秩定理 如果矩陣A有n列，則 A的秩+ A的零空間的維數(shù)=n （回憶第一章r+ p = n,不知道？罰 你看第一章秘籍100遍） r是主元列的個(gè)數(shù)P是自由變炭的個(gè)數(shù)，Ax=0佇多少自 由變磺，就仃多少線性無(wú)關(guān)的基礎(chǔ)解 向吊,也就足說(shuō)A的零空

33、間的維數(shù)是 PP179可逆矩陣性質(zhì)定理續(xù)m.A的列向最形成了的一個(gè)基n.Col A = Rn.o.dim Col A = n.p.rank A = n.q.Nul A = 0r.dim Nul A = 0注：這是因?yàn)锳可逆，A可以初等變 換為單位陣，單位陣地列向眾都線性 無(wú)關(guān)。因?yàn)槌醯茸儞Q不改變線性相關(guān) 性，則說(shuō)明A的n個(gè)列向量也都線性 無(wú)關(guān).Ax=0只有零解。為什么初等變換不改變線性相關(guān) 性？因?yàn)槌醯茸儞Q不改變方程組 Ax=0的解。9/12Chapter 4determinantP187行列式(ij)-cofactor(1嚴(yán)如州P165代數(shù)余子式cofactor expansionP165余

34、因子展開式Theorem 2 det of a triangular matrixIf A is a triangular matrix, then det A is the product of the entries on the main diagonal of AP189定理2三角矩陣的行列式定理三角矩陣的行列式是該矩陣的主對(duì) 角線上元素的乘積。Theorem 3 row operations on determinanta. If a multiple of one row of A is added to another row to produce a matrix B, the

35、n det B =det Ab. If two rows of A are interchanged to produce B.then detB = det Ac. If one row of A is multiplied by k to producedP192定理3矩陣行變換與對(duì)應(yīng)行列式的 值a. 把A的某一行的倍數(shù)加到另一行得到矩陣B.則detB = detAb. 若A的兩行互換得到矩陣B,則 detB = - detAc. 若A的某一行乘以k得到矩陣B,detB = k detAB, then detB = k det ATheorem 4 use determinant to i

36、nvestigate whether matrix is invertibleA square matrix A is invertible iff det A HOP194定理4用行列式判可逆一個(gè)方陣A可逆當(dāng)且僅當(dāng)detA HOTheorem 5 determinant of transpose of AP196定理5轉(zhuǎn)置矩陣的行列式If A is an n x n matrix, then det AT = det A-個(gè)方陣A,它的轉(zhuǎn)宣矩陣的行列式和 它本身的行列式值相等。Theorem 6 Multiplicative PropertyP196定理6矩陣乘法的行列式If A and B

37、 are an n x n matrices, then方陣A和B乘枳的行列式等J： A的行det AB = (det A) (det B)列式乘以B的行列式 det AB = (det A) (det B)Theorem 7 Cramers RuleP201定理7克萊姆法則Let A be an invertible n x n matrix. For any b in設(shè)A是一個(gè)可逆n階方陣，対中任Rn# the unique solution x of Ax = b has entries意向量b,方程組Ax二b的唯一解可用下given by面的方法計(jì)算：det Ai(b)det (b)X

38、idetAXldet AadjugateP203伴隨矩陣Theorem 8 An Inverse Formula定理8逆矩陣計(jì)算公式Let A be an invertible n x n matrix. Then 1adjAA - detA adAVector spaceP215向量空間SubspaceP220子空間Zero SubspaceP220零子空間Subspace spanned by vl.vpP221由向生成（張成）的子空間Null space of an m x n matrix A (written as Nul A)Nul A is a subspace of RnP22

39、6- 227mxn矩陣A的零空間（注意與零子空間 區(qū)別開來(lái)）。Column space of an m x n matrix A (written as Col A)Col A is a subspace of RmP229矩陣A的列空間 記作Col ACol A是R的子空間BasisPivot columns of A form a basis for Col AP238P241基矩陣A的主尤列形成了 Col A的基Coordinates of x relative to the basis BP246向童x相對(duì)于基B的坐標(biāo)Coordinate vector of xP247向量x相對(duì)于基B

40、的坐標(biāo)向AtCoordinate mappingP247坐標(biāo)映射DimensionP256-257維數(shù)Rankrank A + dim Nul A = nP265秩Invertible matrix theoremP267可逆矩陣的秩、維數(shù)定理Change of basisP273基的變換B =bD . , bj, C =given xB(coordinates of vector x relative to the basis B), and bic,., bnc (coordinates of vectors ., bn relative to the basis C|;Then: xc

41、= c-bxbC-B = bjjc,., bnc設(shè) B =bb . # bnh C = CiCnh xb 是 X 相 對(duì)于B上的坐標(biāo)，并且b】c, ., bnc是 基B相對(duì)于C的坐標(biāo).xc= C-Bxb 其中cIb =biC/bncl11/12Chapter 6Eigenvector; EigenvalueP303特征向量；特征值Eigenvectors correspond to distinct eigenvalues對(duì)應(yīng)r不同特征值的特征向量線性無(wú)關(guān)are linearly independentP307n x n matrix A is invertible iff:P312nxn矩陣A是可逆的，當(dāng)且僅當(dāng)：0 is not a