CHAPTER ONE Matrices and System Equations Objective:To provide solvability conditions of a linear equation Ax=b and introduce the Gaussian elimination method, a systematical approach in solving Ax=b, to solve it.

Outline Motivative Example. Elementary row operations and Elementary Matrices. Some Basic Properties of Matrices. Gaussian Elimination for solving Ax=b. Solvability conditions for Ax=b.

Motivative Example (curve fitting) Given three points( )( )( ),find a polynomial of degree 2 passing through the three given points. Solution: Let the polynomial be Where a,b and c are to be determined Ax=b

Question: Why transform to matrix form? To provide a systematic approach and to use computer resource.

Question: How to solve Ax=b systematically? One way is to put Ax=b in triangular form,which can be easily solved by back-substitution.  Definition: A system is said to be in triangular form if in the k - th equation the coefficients of thee first ( k -1) variables are all zero and the coefficient of x k is nonzero ( k = 1, …,n)

Eg1:

Question: How to put Ax=b in triangular form while leaving the solution set invariant? Solution: By elementary row operations as described below.  Definition: Two systems of equations involing the same variables are said to be equivalent if they have the same solution set.

Before introducing elementary operation, we recall some definitions and notations. (§ 1.3) Equality of two matrices. Multiplication of a matrix by a scalar. Matrix addition. Matrix multiplication. Identity matrix. Multiplicative inverse. Nonsingular and singular matrix. Transpose of a matrix.

Def. If and, then the Matrix Multiplication, where. Def. An (n × n) matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I. The matrix B is said to be a multiplicative inverse of A. And B is denoted by A -1. Warning: In general, AB≠BA. Matrix multiplication is not commutative.Definitions

Def. The transpose of an (m × n) matrix A is the (n × m) matrix B defined by for j=1,…,n and i=1,…,m. The transpose of A is denoted by A T. Def. An (n × n) matrix A is said to be symmetric if A T =A. Definitions (cont.)

Some Matrix Properties Let be scalars,A,B and C be matrices with proper dimensions. (Commutative Law) (Associative Law) (Distributive Law)

Some Matrix Properties (cont.)

Notations, The matrix is called an augmented matrix. In general, or.

Moreover,we define

Def: Let and.Then is said to be a linear combination of. Note that.We have the next result. Theorem1.3.1: Ax=b is consistent b can be written as a linear combination of colum vectors of A.

Application 1: Weight Reduction Table 1 Calories Burned Per Hour Weight in lb Exercise Activity Walking 2 mph Running 5.5 mph Bicycling 5.5mph Tennis

Application 1: Weight Reduction (cont.) Table 2 Hours Per Day For Each Activity Exercise schedule walkingRunningBicyclingTennis Monday Tuesday Wednesday Thursday Friday

Application 1: Weight Reduction (end) Solution:

Application 2: Production Costs Table 3 Production Costs Per Item (dollars) Product Expenses ABC Raw materials Labor Overhead and miscellaneous

Application 2: Production Costs (cont.) Table 4 Amount Produced Per Quarter Season Product SummerFallWinterSpring A B C

Application 2: Weight Reduction (cont.) Solution:

Application 2: Weight Reduction (cont.) Solution:

Application 2: Production Costs (end) Table 5 Amount Produced Per Quarter Season SummerFallWinterSpringYear Raw materials1,8702,1602,0701,9608,060 Labor3,4503,9403,8103,58014,780 Overhead and miscellaneous 1,6701,9001,8301,7407,140 Total production cost 6,9908,0007,7107,28029,980 Solution:

Application 5: Networks and Graphs (P.57)

Application 5: Networks and Graphs (cont.) DEF.

Application 5: Networks and Graphs (end) Theorem If A is an n × n adjacency matrix of a graph and represents the ij th entry of A k, then is equal to the number of walks of length from to V i to V j.

Application 6: Information Retrieval (P.59) Suppose that our database, consists of these book titles: B1. Applied Linear Algebra B2. Elementary Linear Algebra B3. Elementary Linear Algebra with Applications B4. Linear Algebra and Its Applications B5. Linear Algebra with Applications B6. Matrix Algebra with Applications B7. Matrix Theory The collection of key words is given by the following alphabetical list ： algebra, application, elementary, linear, matrix, theory

Application 6: Information Retrieval (cont.) Table 8 Array Representation for Database of Linear Algebra Books Books Key Words B1B2B3B4B5B6B7 algebra application elementary linear matrix theory

Application 6: Information Retrieval (end) If the words we are searching for are applied, linear, and algebra, then the database matrix and search vector are given by If we set y= A T x, then

Let’s back to solve Ax=b Eg2

(§ 1.2) Three types of Elementary row operations. I. Interchange two row. II. Multiply a row by. III. Replace a row by its sum with a multiple of another row.

Lead variables and free variables(p.15) Eg:, and are lead variables while and are free variables.

 Def. A matrix is said to be in row echelon form if (i) The first nonzero entry in each row is 1. (ii) If row k does not consist entirely of zero, the number of leading zero entries in row k +1 is grater then the number of leading zero entries in row k. (iii) If there are rows whose entries are all zero, they are below the rows having nonzero entries.  Def. The process of using row operations I, II, and III to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.

 Def. A linear system is said to be overdetermined if there are more equations( m ) than unknowns ( n ). ( m > n ) Warning: Overdetermined systems are usually (but not always) in consistent.  Def. A system of m linear equations in n unknowns is said to be underdetermined if there are fewer equations. (m < n) Overdetermined and Underdetermined

 Def. A matrix is said to be in reduced row echelon form if: (i) The matrix is in row echelon form. (ii) The first nonzero entry in each row is the only nonzero entry in its column.  Def. The process of using elementary row operations to transform a matrix into reduced row echelon form is called Gauss-Jordan reduction. Reduced Row Echelon Form

Application 2: Electrical Networks (P.22)

Application 2: Electrical Networks (end) Kirchhoff’s Laws: 1. At every node the sum of the incoming currents equals the sum of the outgoing currents. 2. Around every closed loop the algebraic sum of the voltage must equal the algebraic sum of the voltage drops.

Application 4: Economic Models For Exchange of Goods (P.25) Exchange of Goods (P.25) FMCFMC 1/2 1/4 F M C 1/3 1/2 1/4

(§ 1.4) Elementary Matrices Type I ( ): Obtained by interchanging rows i and j from identity matrix. Type II ( ): Obtained from identity matrix by multiplying row i with. Type III ( ): Obtained from identity matrix by adding to row j.

means performing type I row operation on A. means performing type II row operation on A. means performing type III row operation on A. means performing type I column operation on A. means performing type II column operation on A. means performing type III column operation on A. Elementary Row / Column Operation

Theorem1.4.2: If E is an elementary matrix, then E is nonsingular and E -1 is an elementary matrix of the same type. With The solution set of a linear equations is invariant under three types row operation.  and have the solution set.

Def. A matrix B is row equivalent to A if there exists a finite sequence of elementary matrices such that Row Equivalent (P.71) Theorem1.4.3 (a) A is nonsingular. (b) Ax=0 has only the trivial solution 0. (c) A is row equivalent to I.

(a) (b) Let be a solution of Ax=0. (b) (c) Let A ~ U, where U is in reduced row echelon form. Suppose U contains a zero row. by Th 1.2.1, Ux=0 has a nontrivial solution thus A~I. (c) (a) A~I  A= E 1 …… E k for some E 1 … E k ∵ each E i is nonsingular. ∴ A is nonsingular. (by Th.1.2.1) row Proof of Theorem 1.4.3

Corollary1.4.4 Ax=b has a unique solution A is nonsingular. Pf: " “ The unique solution is. " " Suppose is the unique solution and A is singular. is also a solution of Ax=b. A is nonsingular.

BUT in general, and AB=AC B=C. Eg. Moreover,AC=AB while.

If A is nonsingular and row equivalent to I, so there exists elementary matrices such that then, E k … E 1 (A | I)= ( E k … E 1 ‧ A | E k … E 1 ‧ I) E k … E 1 (A | I)= ( E k … E 1 ‧ A | E k … E 1 ‧ I) ( by ) = (I | E k … E 1 ‧ I) = (I | E k … E 1 ‧ I) ( by ) = (I | A -1 ) = (I | A -1 ) Method For Computing

Q: Compute A -1 if. Sol: Example 4. (P.73)

Q: Compute A -1 if. Sol: Example 4. (cont.)

Diagonal and Triangular Matrices Def. An n × n matrix A is said to be upper triangular if a ij =0 for i > j and lower triangular if a ij =0 for i > j. Def. An n × n matrix B is diagonal if a ij =0 whenever i ≠ j. Triangular Factorization If an n × n matrix C can be reduced to upper triangular form using only row operation III, then C has an LU factorization. The matrix L is unit lower triangular, and if i > j, then l ij is the multiple of t he j th row subtracted from the i th row during the reduction process.

Example 6. (P.74) row operation III Mark: