线性代数英文讲义课件.ppt

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1、Chapter 1,Matrices and Systems of Equations,Systems of Linear Equations,Where the aijs and bis are all real numbers, xis are variables . We will refer to systems of the form (1) as mn linear systems.,Definition Inconsistent : A linear system has no solution.Consistent : A linear system has at least

2、one solution.,Example() x1 + x2 = 2 x1 x2 = 2,() x1 + x2 = 2 x1 + x2 =1,() x1 + x2 = 2 x1 x2 =-2,Definition Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.,Three Operations that can be used on a system to obtain an equivalent system

3、:,. The order in which any two equations are written may be interchanged.,. Both sides of an equation may be multiplied by the same nonzero real number.,. A multiple of one equation may be added to (or subtracted from) another.,nn Systems,Definition A system is said to be in strict triangular form i

4、f in the kthequation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, ,n).,is in strict triangular form.,Example The system,Example Solve the system,Elementary Row Operations:. Interchange two rows. Multiply a row by a nonzero real number. Replace a

5、 row by its sum with a multiple of another row.,Example Solve the system,2 Row Echelon Form,pivotal row,pivotal row,Definition A matrix is said to be in row echelon form. If the first nonzero entry in each nonzero row is 1. If row k does not consist entirely of zeros, the number of leading zero entr

6、ies in row k+1 is greater than the number of leading zero entries in row k. If there are rows whose entries are all zero, they are below the rows having nonzero entries.,Example Determine whether the following matrices arein row echelon form or not.,Definition The process of using operations , , to

7、transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.,Definition A linear system is said to be overdetermined if there are more equations than unknows.A system of m linear equations in n unknows is said to be underdetermined if there are few

8、er equations than unknows (mn).,Example,Definition A matrix is said to be in reduced row echelon form if:. The matrix is in row echelon form. The first nonzero entry in each row is the only nonzero entry in its column.,Homogeneous SystemsA system of linear equations is said to be homogeneous if the

9、constants on the right-hand side are all zero.,Theorem 1.2.1 An mn homogeneous system of linear equations has a nontrivial solution if nm.,3 Matrix Algebra,Matrix Notation,Vectorsrow vector,column vector,1n matrix,n1 matrix,Definition Two mn matrices A and B are said to be equal if aij=bij for each

10、i and j.,Scalar MultiplicationIf A is a matrix and k is a scalar, then kA is the matrix formed by multiplying each of the entries of A by k.,Definition If A is an mn matrix and k is a scalar, then kA is the mn matrix whose (i, j) entry is kaij.,Matrix AdditionTwo matrices with the same dimensions ca

11、n be addedby adding their corresponding entries.,Definition If A=(aij) and B=(bij) are both mn matrices,then the sum A+B is the mn matrix whose (i, j) entry is aij+bij for each ordered pair (i, j).,Example,Let,Then calculate,。,Matrix Multiplication,Definition If A=(aij) is an mn matrix and B=(bij) i

12、s an nr matrix, then the product AB=C=(cij) is the mr matrix whose entries are defined by,Example,then calculate AB.,1. If,2. If,then calculate AB and BA.,Matrix Multiplication and Linear SystemsCase 1 One equation in Several UnknowsIf we let and then we define the product AX by,Case 2 M equations i

13、n N UnknowsIf we let and then we define the product AX by,Definition If a1, a2, , an are vectors in Rm and c1, c2, , cn are scalars, then a sum of the form c1a1+c2a2+cnan is said to be a linear combination of the vectors a1, a2, , an .,Theorem 1.3.1 (Consistency Theorem for Linear Systems)A linear s

14、ystem AX=b is consistent if and only if b can be written as a linear combination of the column vectors of A.,Theorem 1.3.2 Each of the following statements is valid for any scalars k and l and for any matrices A, B and C for which the indicated operations are defined. A+B=B+A (A+B)+C=A+(B+C) (AB)C=A

15、(BC) A(B+C)=AB+AC (A+B)+C=AC+BC (kl)A=k(lA) k(AB)=(kA)B=A(kB) (k+l)A=kA+lA k(A+B)=kA+kB,The Identity Matrix,Definition The nn identity is the matrix where,Matrix Inversion,Definition An nn matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I.Then matrix B is

16、said to be a multiplicative inverse of A.,Definition An nn matrix is said to be singular if it does not have a multiplicative inverse.,Theorem 1.3.3 If A and B are nonsingular nn matrices, then AB is also nonsingular and (AB)-1=B-1A-1,The Transpose of a Matrix,Definition The transpose of an mn matri

17、x A is the nm matrix B defined by bji=aij for j=1, , n and i=1, , m. The transpose of A is denoted by AT.,Algebra Rules for Transpose:(AT)T=A (kA)T=kAT (A+B)T=AT+BT (AB)T=BTAT,Definition An nn matrix A is said to be symmetric if AT=A.,4. Elementary Matrices,If we start with the identity matrix I and

18、 then perform exactly one elementary row operation, the resulting matrix is called an elementary matrix.,Type I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I.,Example Let,and let A be a 33 matrix,then,Type II. An elementary matrix of type II is a matrix obtained

19、 by multiplying a row of I by a nonzero constant.,Example Let,and let A be a 33 matrix,then,Type III. An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.,Example Let,and let A be a 33 matrix,In general, suppose that E is an nn elementary matri

20、x. E is obtained by either a row operation or a column operation. If A is an nr matrix, premultiplying A by E has the effect of performing that same row operation on A. If B is an mn matrix, postmultiplying B by E is equivalentto performing that same column operation on B.,Example Let , Find the ele

21、mentary matrices ， ，such that .,Theorem 1.4.1 If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.,Definition A matrix B is row equivalent to A if there exists a finite sequence E1, E2, , Ek of elementary matrices such that B=EkEk-1E1A,Theorem 1.4.2 (

22、Equivalent Conditions for Nonsingularity)Let A be an nn matrix. The following are equivalent: A is nonsingular. Ax=0 has only the trivial solution 0. A is row equivalent to I.,Theorem 1.4.3 The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.,If

23、A is nonsingular, then A is row equivalent to I and hence there exist elementary matrices E1, , Ek suchthat EkEk-1E1A=I multiplying both sides of this equation on the right by A-1 EkEk-1E1I=A-1,Thus (A I),(I A-1),row operations,A method for finding the inverse of a matrix,Example Compute A-1 if,Exam

24、ple Solve the system,Diagonal and Triangular Matrices,An nn matrix A is said to be upper triangular if aij=0 for ij and lower triangular if aij=0 for ij.,An nn matrix A is said to be diagonal if aij=0 whenever ij .,A is said to be triangular if it is either upper triangular or lower triangular.,5. P

25、artitioned Matrices,C=,-2 4 1 3 1 1 1 1 3 2 -1 24 6 2 2 4,C11 C12 = C21 C22,-1 2 1B= 2 3 1 1 4 1,=(b1, b2, b3),AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3),In general, if A is an mn matrix and B is an nr that hasbeen partitioned into columns (b1, , br), then the blockmultiplication of A times B is given by AB=(

26、Ab1, Ab2, , Abr),If we partition A into rows, then,Then the product AB can be partitioned into rows as follows:,Block Multiplication,Let A be an mn matrix and B an nr matrix.,Case 1 B=(B1 B2), where B1 is an nt matrix and B2is an n(r-t) matrix.,AB= A(b1, , bt, bt+1, , br) = (Ab1, , Abt, Abt+1, , Abr

27、) = (A(b1, , bt), A(bt+1, , br) = (AB1 AB2),Case 2 A= ,where A1 is a kn matrix and A2is an (m-k)n matrix.,Thus,Case 3 A=(A1 A2) and B= , where A1 is an ms matrixand A2 is an m(n-s) matrix, B1 is an sr matrix and B2 is an (n-s)r matrix.,Thus,Case 4 Let A and B both be partitioned as follows：,A11 A12

28、kA= A21 A22 m-k s n-s,B11 B12 sB= B21 B22 n-s t r-t,Then,In general, if the blocks have the proper dimensions, the block multiplication can be carried out in the same manner as ordinary matrix multiplication.,Example,Let,Then calculate AB.,Example,Let A be an nn matrix of the form,where A11 is a kk matrix (kn). Show thatA is nonsingular if and only if A11 and A22are nonsingular.,