线性代数及其应用术语要点中英对照

1、Chapter 1-Review1. 线性方程组 Systems of Linear Equations (Linear System)P3 】关键词：coefficient 系数P2; constant term 常数(项)讲5C-P1； linear equation 线性方程P2； variable 未知数(或变元)有个方程个未知数仗讣)的姣性方程组可表示为：1) + a2 + + 知片=(/i.a b 为 m 维列向)3) Ax=b (A是mx 矩阵；x tb为m维列向)4) Augmented “tairix(増 广矩阵)-(其中第/a j勻列是变元叼的系数)2线性方程组解的憎况(S

2、olution Status)【P41)2)No solution无解Has Solution 有解a) Exactly one solution (unu/ne sohitioit)b) Infinite many sobitions无穷多解唯一解3阶梯形(Echelon Forms)P14关键leading entry 先导元素P14; pivot position 主元位置P16;1) 3 conditions of echelon form matrix阶梯形矩阵的三个条件(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 导元素都在上一行先导元素的右边c) In each colunui, an entry below the leading entry is 0与先导元素同列且在其下部的元素全 为02) 2 additional conditions of Reduced Echelon Forms 简化阶梯形的额外两个性质：a) Tlie leading entry of each nonzero zw is

4、1每一非零行的先导元素都是 1b) Each leading 1 is the ONLY nonzero entry of its column先导元素是其所在列唯一非零元素注：与线性方程组结合： f . f No free variables unique solution free variable * infinitely many solutionsfree variable至少有一个自由变注：结合简化阶梯形采用反证法轻松搞定！Additionally,此外：if r = #pivot positions, p = free variables, n =体variables then

5、r+P = n,#0 - number of Q (J 的个数)6. 非齐次线性方程组解的结构定理(Structure of Solution Set ofP53Nonhomogeneous System)关键词：nonhomogeneous system非齐次线性方程组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如果是非齐次线性方程组Ax = b的一个解，H是对应齐次线性方程组M = 0的通解。(XU二 0也称为Ax = b的导出组)则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 为标)有解三 b 是 aba.an 的线性组合8. 线性无关/ 相关(Linear Independent / Dependent)P65关键词trivial solutions非零解/非平凡解P51；莎m维空间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)如果方程组+%刘+.+*”= 0只有零解(.v2.v2 .x全是0),则心， a2,.anttt 性无关。Vector set aj, 2, “幵石 linear independent tfxjaj + + xn= 0 if .xn are not all 0.若方程组xiaj+x2fl2+有非零解(xjx2 .X.,不全是O),则向量组“ a2,.an线性相关。2) Theorem 7 Characterization of Linearly Dependent定理7线性相关和线性组合的关系定理 P68Vector set qi, a2

9、,.an) is linear dependent Exist vector a2(l i h), which is a linear conibiiialioii of the other vectors向组(alfa2f.an线性相关存在某向量心(区)是其它向的线性组合注：由线性相关定义Xjaj +x申：+ . +%”= 0 , xjx2 . xn不全是0则线性相关。设X： H 0(7 0 把移到等式另一边yfli = - ( Xfi +七a： + . + xffln ),然后两边除以Xj (因为忑H 0 )即得证向量al i m r (1 i /*), which is a linear

10、 combination of the other vectors如果向组中向个数大于向的维数皿，则向组线性相关。注：不知如何证明？看本表第5项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线性相关汪：还是不知如何证明？看本格上面的定义100 JB o9. 等价定理(Theorem。P43关键词：Rr,m 维空间P28; subset of Rrn spanned (or generated) by % j 由

11、 v.vp 张成(或生成的)的的子空间P35;1) For each b in 时；the system Ax = b has a solution对于 对 中的每一个向量b,线性方程组4丫 = b都有一个解2) Each b in P “ a linear combination of die cobinuis of A. 时中的毎一个向 b 都是矩阵 A 的 列向量的线性组合3) The coliinuis of A sp(ui - 矩阵 A 的列向生成4) The matrix A has a pivot position in every矩阵A毎一行都有一个主元位注：。3)根据定义显然

12、成立；4)可用定理2采用反证法10. 补充齐次方程组基础解系定理(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 (为什么,看本衣第S项)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矩阵代数matrix operationsPIO 刀矩阵的运算main diagonal of matrixPIO 刀矩阵的主对角线diagonal matrixPIO 刀对角矩阵identity matrix lnP45+ PIO刀n x n单位矩阵matrix additionPIO 刀矩阵加法scalar multiplicationP109数乘（矩阵）matrix multiplicationP109矩阵乘法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矩阵B: n x p矩阵，矩阵的各列向量为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矩阵AB的第j列Vj都是A的所有列 向量血的线性组合.（其中各个 权是B中对应列匕的元素）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矩阵运算规则A: m x n矩阵B, C:在每行中，尺寸都符合那行加 法和乘法定义的矩阵k, t：标量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）矩阵加法和数乘2) Multiplication A|BC| = (AB)C A(B+C) = AB +AC (B+CJA = BA +CA k(AB|-(kA)B = A(kB) UA = A = Alm2）矩阵乘法commuteP113可交换（矩阵乘法）Warnings:In general AB H BA AB=AC 壬 B

18、= CAB=O 丰 A - 0 or B - 0P114transpose of a matrixPH5矩阵的转置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（矩阵）

19、可逆的matrix inverseP119矩阵的逆singular matrixP119奇异矩阵nonsingular matrixP119非奇异矩阵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)P119二阶方阵人=:3 可逆的充要条件 ad-bc H 0或记作|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系数为n阶可逆方阵A的线 性方程组Ax=b的解的情况定理若A是一个n阶可逆矩阵，那么对于 n维空间R”中的每一个列向量b方 程组Ax = b都有唯一解x = A 2bTheorem 6 Rules ofA, B: n x n invertible matricesP121定理6矩阵的逆运算规则(A1)1 = A (AB)1 = BW IaT

21、= (A1)7elementary matrixP122初等矩阵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左乘初等矩阵等价于进行一次与初等矩阵一样的行初等 变换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可逆矩阵判断定理一个mm矩阵A是可逆的当且仅当 A行等价于In（就是说A可以行化 简成人）。并且，在这种情况下，任 何一系列把A行化简成厶的操作， 都可以把人转化成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用初等行变换求逆矩阵：把增广矩阵A | I化简，如果

24、A行等 价于单位阵,则A | I能化简成I | A1,否则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可逆矩阵性质定理I、列断言等价a. A是可逆的b. A行等价J：一个n阶单

25、位阵。c. A有n个主尤位置。d. 矩阵方程Ax = 0仅有平凡解（零f. The linear transformation x i Ax isf. 线性变换XI Ax是一对一的。one-to-one.g. The equation Ax= b has only one solution forg-对珅中任意的一个向星b,矩阵方each b in Rn.程Ax= b有唯一解。h. The columns of A span Rn.h.A的列张成IT.i. The linear transformation x i Ax mapsI.线性变换X Ax把R”映射到RSRn to Rn.J-存在一

26、个n x n矩阵C使CA =1.j. There is an n x n matrix C such that CA = I.k.存在一个n x n矩阵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分块矩阵multiplication of partitioned matricesP135分块矩阵的乘法Partitions of A and B should be conformabl

27、e forP136A和B的分块矩阵要相乘的话，Ablock multiplication和B的分法应遵从矩阵乘法定义The column partition of A matches the rowA的列分法应与B的行分法一致partition of B（左边大小列=右边大小行）Theorem 10 column-row expansion of ABP137定理10 AB乘法的列行展开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的列形成一个线性无关集。AB = coli(A) col2(A) . coln(A)row2(F) rown(BcoljfA) roWtfB) +.+ coln(A) rown(B)subspaceP168子空间column space of AColA = all linear combinations of the columns of A =kjBi + . + knan (ki(1in)6R)P169A的列牢间ColA = A的所有列的线性组合形成的 向最的集合null space of ANul A = all solutions to the homogen

29、eous equationAx =0P169A的零空间Nul A =齐次线性方程组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的零空间定理mxn矩阵A的零空间是R”的子空间 （这是因为Ax = 0的解向最是n维 的，所以它是n维空间的了空间）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.也就是说，有着m个方程n个未知数 的方程组Ax=0的通解是R的子空间.basisP170基8/12P172 A的列空间定理A的主元列形成了 A的列空间的一个基。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相对J * B的坐标向量（对照解析几何中，相对J：x轴,y轴,z 轴的坐标）P177子空间的维数非零子空间的维数，用dimH衷示， 它是H的任意一个甚中，向灵的个 数。零子空间的维数定义成0（注意：与向量的维数区别！）P178秩定理14矩阵的秩定理 如果矩阵A有n列，则 A的秩+ A的零空间的维数=n （回忆第一章r+ p = n,不知道？罚 你看第一章秘籍100遍） r是主元列的个数P是自由变炭的个数，Ax=0佇多少自 由变磺，就仃多少线性无关的基础解 向吊,也就足说A的零空

33、间的维数是 PP179可逆矩阵性质定理续m.A的列向最形成了的一个基n.Col A = Rn.o.dim Col A = n.p.rank A = n.q.Nul A = 0r.dim Nul A = 0注：这是因为A可逆，A可以初等变 换为单位阵，单位阵地列向眾都线性 无关。因为初等变换不改变线性相关 性，则说明A的n个列向量也都线性 无关.Ax=0只有零解。为什么初等变换不改变线性相关 性？因为初等变换不改变方程组 Ax=0的解。9/12Chapter 4determinantP187行列式(ij)-cofactor(1严如州P165代数余子式cofactor expansionP165余

34、因子展开式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2三角矩阵的行列式定理三角矩阵的行列式是该矩阵的主对 角线上元素的乘积。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矩阵行变换与对应行列式的 值a. 把A的某一行的倍数加到另一行得到矩阵B.则detB = detAb. 若A的两行互换得到矩阵B,则 detB = - detAc. 若A的某一行乘以k得到矩阵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用行列式判可逆一个方阵A可逆当且仅当detA HOTheorem 5 determinant of transpose of AP196定理5转置矩阵的行列式If A is an n x n matrix, then det AT = det A-个方阵A,它的转宣矩阵的行列式和 它本身的行列式值相等。Theorem 6 Multiplicative PropertyP196定理6矩阵乘法的行列式If A and B

37、 are an n x n matrices, then方阵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克莱姆法则Let A be an invertible n x n matrix. For any b in设A是一个可逆n阶方阵，対中任Rn# the unique solution x of Ax = b has entries意向量b,方程组Ax二b的唯一解可用下given by面的方法计算：det Ai(b)det (b)X

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

39、6- 227mxn矩阵A的零空间（注意与零子空间 区别开来）。Column space of an m x n matrix A (written as Col A)Col A is a subspace of RmP229矩阵A的列空间 记作Col ACol A是R的子空间BasisPivot columns of A form a basis for Col AP238P241基矩阵A的主尤列形成了 Col A的基Coordinates of x relative to the basis BP246向童x相对于基B的坐标Coordinate vector of xP247向量x相对于基B

40、的坐标向AtCoordinate mappingP247坐标映射DimensionP256-257维数Rankrank A + dim Nul A = nP265秩Invertible matrix theoremP267可逆矩阵的秩、维数定理Change of basisP273基的变换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设 B =bb . # bnh C = CiCnh xb 是 X 相 对于B上的坐标，并且b】c, ., bnc是 基B相对于C的坐标.xc= C-Bxb 其中cIb =biC/bncl11/12Chapter 6Eigenvector; EigenvalueP303特征向量；特征值Eigenvectors correspond to distinct eigenvalues对应r不同特征值的特征向量线性无关are linearly independentP307n x n matrix A is invertible iff:P312nxn矩阵A是可逆的，当且仅当：0 is not a