#### lec09

Berkeley, MATH 110
Excerpt: ... MATH 110 Lecture Notes 9 GSI Carter July 7, 2008 1 Elementary Matrices 1. interchanging two rows 2. multiplying a row by a nonzero scalar 3. adding a multiple of a row to some other row There are three types of elementary row operations : We can show that these operations are performed by multiplication on the left by a square matrix using the following characterization of matrix multiplication: A(b1 | b2 | | bn ) = (Ab1 | Ab2 | | Abn ). There are corresponding columns operations which are performed by multiplication on the right, which we can show using the fact that (AB)t = B t At . All elementary matrices are invertible. 1 ...

#### 09S-60-Lecture06

Cal Poly Pomona, SG 064747
Excerpt: ... Row Operations Recall that the following three operations on a matrix are called elementary row operations : (i) multipling a row through by a nonzero constant, (ii) interchanging two rows, 4 MATH 60 SPRING 2009 (iii) adding a multiple of one row to another. Denition. An n n matrix is called an elementary matrix if it can be obtained from the n n identity matrix In by performing a single elementary row operation. Since an elementary row operation can always be undone, it follows that: Theorem 1. Every elementary matrix is invertible. We will focus now (just to get some practice with the concepts) on 2 2 elementary matrices. These are 2 2 matrices that result from performing a single elementary row operation on the 2 2 identity matrix. Since we are dealing with 2 2 matrices, there are not too many possibilities. Every 2 2 elementary matrix is of one of the following forms: 1 k 0 , 1 1 0 k , 1 0 1 1 , 0 k 0 0 , 1 1 0 0 , k where k is a nonzero real number. ...

#### Math221Lecture001BHandouts

UMBC, MATH 221
Excerpt: ... of the variables in the system each row contains the coefcients of the variables in one equation the right most column contains the right hand-sides of the equations in the system Clint Lee Math 221 Lecture 1: Solving Systems of Linear Equations 8/14 The Augmented Matrix continued For a system of m equations in n unknowns the augmented matrix has m rows and n + 1 columns, that is, it is an m n + 1 matrix (rows columns). The Augmented Matrix for a System of m Equations in n Unknowns a11 a22 . . . a12 a22 . . . . . a1n a2n . . . b1 b2 . . . The portion of the augmented matrix containing the coefcients of the variables is the . It is an matrix. This is everything to the left of the vertical line in the matrix above. Clint Lee Math 221 Lecture 1: Solving Systems of Linear Equations 9/14 Elementary Row Operations The augmented matrix for a linear system is replaced by the augmented matrix an equivalent system using: Th ...

#### MAS3105hw1

CUNY Baruch, MAS 3105
Excerpt: ... Linear Algebra Spring 06 Homework 1 Due Monday, February 6 at the beginning of class. No late homework will be accepted. Name: This homework is worth 30 points. For credit, show all your work in a clear and organized manner. No credit will be given if you just write the answer. 1. 1, page 88. Note: solve the system by transforming the augmented matrix into reduced row echelon form. No credit will be given for using other methods. Clearly indicate the elementary row operations that you perform at each step. 2. 5, page 89. 3. 5, (f) and (i) page 26. Clearly indicate the elementary row operations that you perform at each step. 4. 17, page 59. 5. 18, page 59. 6. 22, page 60. 7. 25, page 60. 8. 10 (g) and (h), page 70. Clearly indicate the elementary row operations that you perform at each step. 9. 15, page 71. 10. 17, page 71. 11. 18, page 71. 1 ...

#### CPSC5115_Ch06

Columbus State University, CPSC 5115
Excerpt: ... olve this set of equations, but first it is important to discuss what the mathematicians call elementary row operations . The student should note that there are also elementary column operations, but that these are of less use in our studies. There are three elementary row operations . 1) Exchanging two rows in a matrix, and 2) Multiplying a row by a non-zero constant, and 3) Replacing a row by the sum of that row and a non-zero multiple of another row. Note that the third operation specifies the non-zero product of a row, as adding a zero product of one row to another row does not change the matrix. The insight of the method of elementary row operations is that in some sense they do not change the matrix. For example, the solution to a set of linear equations does not depend on what equation is written first; thus swapping rows should not change the solution. Before working with this sample matrix and its associated set of equations, let us consider two related sets of equations. Each set of equations is relat ...

#### Lecture7

Lehigh, IE 316
Excerpt: ... operations, denoted O(m3). To take the inner product of two n-dimensional vectors takes O(n) operations (n multiplications and n additions). Hence, each iteration of the naive implementation of the Simplex method takes O(m3 + mn) iterations. We'll try to improve up on this. IE316 Lecture 7 11 Improving the Efficiency of Simplex Again, the matrix B -1 plays a central role in the simplex method. If we had B -1 available at the beginning of each iteration, we could easily compute everything we need. Recall that B changes in only one column during each iteration. How does B -1 change? It may change a lot, but we can update it instead of recomputing it. IE316 Lecture 7 12 Way Back in Linear Algebra Recall from linear algebra how to invert a matrix by hand. We use elementary row operations . An elementary row operation is adding a multiple of a row to the same or another row. To invert a matrix, we use elementary row operation to change the matrix into the identity. We ...

#### Lecture7

Lehigh, IE 406
Excerpt: ... order of m3 operations, denoted O(m3). To take the inner product of two n-dimensional vectors takes O(n) operations (n multiplications and n additions). Hence, each iteration of the naive implementation of the Simplex method takes O(m3 + mn) operations. We'll try to improve upon this. IE406 Lecture 7 11 Improving the Efficiency of Simplex Again, the matrix B -1 plays a central role in the simplex method. If we had B -1 available at the beginning of each iteration, we could easily compute everything we need. Recall that B changes in only one column during each iteration. How does B -1 change? It may change a lot, but we can update it instead of recomputing it. IE406 Lecture 7 12 Way Back in Linear Algebra Recall from linear algebra how to invert a matrix by hand. We use elementary row operations . An elementary row operation is adding a multiple of a row to the same or another row. To invert a matrix, we use elementary row operation to change the matrix into the ide ...

#### 13echelon

ASU, MAT 342
Excerpt: ... an elementary matrix of type 1. 2. If R replaces row 3 by the sum of row 3 and 1 0 0 0 1 0 R(I5 ) = 4 0 1 0 0 0 0 0 0 is an elementary matrix of type 2. 3. If R switches rows 2 and 4, then 1 0 R(I5 ) = 0 0 0 is an elementary matrix of type 3. 0 0 0 1 0 0 0 1 0 0 4 times row 1, then 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 Proposition. Let R be an elementary row operation, and let E be the corresponding elementary matrix. Then for all A Mmn we have R(A) = EA. Proof. In each case this follows easily from the properties of matrix multiplication. QED Corollary. Every elementary matrix is invertible. In fact, if E is an elementary matrix of type t, then E 1 is also an elementary matrix of type t. Proof. The corresponding properties of elementary row operations are obvious: 1. If R multiplies row i by c, then R1 multiplies row i by c1 ; 2. If R replaces row i by the sum of row i and c times row j, then R1 repla ...

#### ex4

University of Florida , STA 6329
Excerpt: ... STA6329 Exercise 4 Due September 23 (Tuesday, Before class) 1. In class we have seen an example that a matrix is transferred a matrix with a 0 row with elementary row operations . More precisely, suppose by a sequence of elementary row operations E 1 , E 2 ,.,E n , we can transfer a square matrix A into a matrix B which has a row of all zeros, i.e., E n E 2 E 1 A = B, where B has a 0 row. Show that there is no way we can find another sequence of elementary row operations to transfer A into the identity matrix. 2. Problem 1 of Chapter 3 (p.26). 3. Problem 3 of Chapter 4 (p.46). Please change the word "matrix" to "vector" . B) R Exercises: (Hand in R program and outputs.) 4. Plot the data "SMOKING" in the data set and add the regression line. Let the smoking rate be the x-axis and lung cancer rate be the y-axis. Give a title and descriptions at the two axes. 5. Plot the 3-D picture for f (x, y) = (x sin(20y)+y sin(20x)2 cosh(sin(10x)x)+(x sin(10y)-y sin(10x)2 cosh(cos(20y)y) for x [-1, 1], y [0, 1]. ...

#### Corrections

N.C. State, MA 405
Excerpt: ... Corrections to class notes on October 10th, 2006 MA405 Fall 2006 In the end of class I made a mistake on the board. The following are the corrected versions of the Theorems stated in class about the row and column equivalence of matrices. Please correct your lecture notes! Theorem 1. Let A, B Rmn . Then the following are equivalent: 1. R(AT ) = R(B T ) 2. N (A) = N (B) 3. A is row equivalent to B 4. B can be obtained from A via elementary row operations 5. There exists P Rmm non-singular matrix such that B = P A. 6. EA = EB Theorem 2. Let A, B Rmn . Then the following are equivalent: 1. R(A) = R(B) 2. N (AT ) = N (B T ) 3. A is column equivalent to B 4. B can be obtained from A via elementary column operations 5. There exists Q Rnn non-singular matrix such that B = AQ. 6. EAT = EB T ...

#### ps3-M340LFall2007

University of Texas, M 340
Excerpt: ... eft multiplication by E1 performs this row operation on X.) 1 2 Let E2 := R22 . What row operation does left multiplication by E2 perform on a 2 n matrix? 0 1 (Now, an important observation: row operations can be achieved by left multiplication by square matrices.) 2 3 Let Z = . Compute E2 Z and ZE2 . 4 5 (Note that E2 Z = ZE2 unlike multiplication of numbers! This is the reason why we need to distinguish between left multiplication by a matrix and right multiplication by a matrix.) What is the inverse row operation of the elementary row operation achieved by left multiplication by E2 ? Here, inverse row operation refers to the row operation that will undo the eect of left multiplication by E2 . Let F2 R22 be the matrix such that left multipliction by F2 achieves this inverse row operation. Find F2 . Compute E2 F2 and F2 E2 . What row operation does left multiplication by E2 F2 achieve? (Recall that elementary row operations are reversible. This i ...

#### ps3-M340LFall2007

University of Texas, M 340
Excerpt: ... eft multiplication by E1 performs this row operation on X.) 1 2 Let E2 := R22 . What row operation does left multiplication by E2 perform on a 2 n matrix? 0 1 (Now, an important observation: row operations can be achieved by left multiplication by square matrices.) 2 3 Let Z = . Compute E2 Z and ZE2 . 4 5 (Note that E2 Z = ZE2 unlike multiplication of numbers! This is the reason why we need to distinguish between left multiplication by a matrix and right multiplication by a matrix.) What is the inverse row operation of the elementary row operation achieved by left multiplication by E2 ? Here, inverse row operation refers to the row operation that will undo the eect of left multiplication by E2 . Let F2 R22 be the matrix such that left multipliction by F2 achieves this inverse row operation. Find F2 . Compute E2 F2 and F2 E2 . What row operation does left multiplication by E2 F2 achieve? (Recall that elementary row operations are reversible. This i ...

#### 9-C Solving Linear Eqns

Dallas, CPB 021000
Excerpt: ... quations by performing a sequence of the following elementary row operations on the augmented matrix: Elementary Row Operations I. Interchange two rows. II. Multiply one row by a nonzero number. III. Add a multiple of one row to a dierent row. Do you see how we are manipulating the system of linear equations by applying each of these operations? When a sequence of elementary row operations is performed on an augmented matrix, the linear system that corresponds to the resulting augmented matrix is equivalent to the original system. That is, the resulting system has the same solution set as the original system. Our strategy in solving linear systems, therefore, is to take an augmented matrix for a system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained. In particular, we bring the augmented matrix to Row-Echelon Form: Row-Echelon Form A matrix is said to be in row-echelon form if 1. All rows consisting entirely ...

#### Alg_TuteWk9_07

Allan Hancock College, MAT 1102
Excerpt: ... y-z 3x + 5y - z x + 5y + 3z = = = 0 10 20 (5) (6) (7) 2 Using elementary row operations to reduce the augmented matrix for this system to reduced row echelon form. Work systematically from left to right, using the diagonal entry in each column to clear that column. Write out the parametric form of the solutions. In the Lab: 1. Refer to questions 3 and 5 above. Enter the augmented matrix A for the system of equations in each case, and use MATLAB to row-reduce it. Interpret the output clearly in each case. Note that if we enter a matrix A in square brackets as follows, A = [ 1 2 -1 1; 0 1 3 3; 0 0 2 4], then the command rref(A) gives its reduced row-echelon form. Also try rrefmovie(A), where you can control the pace and see each column being cleared systematically from left to right. 3) Commands: 5) Commands: Output matrix: Output matrix: Conclusion: Conclusion: Show that the planes given in Q5 intersect in a line: ie write your solution for 5) in a form that shows that it is the vector equation of a ...

#### AlgTute6_04

Allan Hancock College, MAT 1102
Excerpt: ... to check if any are perpendicular. x+y-z 3x + 5y - z x + 5y + 3z = = = 0 10 20 (3) (4) (5) 2 Using elementary row operations to reduce the augmented matrix for this system to reduced row echelon form. Work systematically from left to right, using the diagonal entry in each column to clear that column. Write out the parametric form of the solutions. In the Lab: 1. Refer to questions 3 and 5 above. Enter the augmented matrix A for the system of equations in each case, and use MATLAB to row-reduce it. Interpret the output clearly in each case. Note that if we enter a matrix A in square brackets as follows, A = [ 1 2 -1 1; 0 1 3 3; 0 0 2 4], then the command rref(A) gives its reduced row-echelon form. Also try rrefmovie(A), where you can control the pace and see each column being cleared systematically from left to right. 3) Commands: 5) Commands: Output matrix: Output matrix: Conclusion: Conclusion: Show that the output for the system in 5) yields a line of intersection given by (x, y, z) = (1 + 2t, ...

#### helper_9-9-05

BU, MATH 242
Excerpt: ... MA 242 Solving systems of linear equations Consider the linear system of equations 2x1 + x2 - x3 = 6 x1 + x2 = 3 x1 + x3 = 1. September 9, 2005 1 MA 242 Let's do a two-variable example more systematically: 3x + y = -2 -x + 3y = 4 September 9, 2005 2 MA 242 Elementary row operations on a matrix September 9, 2005 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Replace a row by a nonzero multiple of itself. Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. (Note that row equivalence is an equivalence relation.) Theorem. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. Now let's return to the original 3-variable example and systematically use row operations: 2x1 + x2 - x3 = 6 x1 + x2 = 3 x1 + x3 = 1. 3 MA 242 September 9, 2005 4 ...

#### helper_9-7-07

BU, MATH 242
Excerpt: ... MA 242 Fractal examples: Consider the square S = {(x, y) | 0 x 1, 0 y 1} and three dierent ways to map S inside of itself. September 7, 2007 1 MA 242 Solving systems of linear equations Consider the linear system of equations 2x1 + x2 x3 = 6 x1 + x2 = 3 x1 + x3 = 1. September 7, 2007 2 MA 242 Lets do a two-variable example more systematically: 3x + y = 2 x + 3y = 4 September 7, 2007 3 MA 242 Elementary row operations on a matrix September 7, 2007 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Replace a row by a nonzero multiple of itself. Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. (Note that row equivalence is an equivalence relation.) Theorem. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. Now lets return to the or ...

#### helper_9-8-06

BU, MATH 242
Excerpt: ... MA 242 Fractal examples: Consider the square S = {(x, y) | 0 x 1, 0 y 1} and three different ways to "map" S inside of itself. September 8, 2006 1 MA 242 Solving systems of linear equations Consider the linear system of equations 2x1 + x2 - x3 = 6 x1 + x2 = 3 x1 + x3 = 1. September 8, 2006 2 MA 242 Let's do a two-variable example more systematically: 3x + y = -2 -x + 3y = 4 September 8, 2006 3 MA 242 Elementary row operations on a matrix September 8, 2006 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Replace a row by a nonzero multiple of itself. Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. (Note that row equivalence is an equivalence relation.) Theorem. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. Now let's return to the original 3-variable example a ...

#### Linear-Preview-01

Excerpt: ... Preview Questions Topic # 01 : Systems of Linear Equations Chapter 1, Section 1 Sample Questions 1. Give an example of an equation that is not linear. 2. How many solutions can a set of two linear equations have? 3. What is. a matrix? 4. When specifying the size of a matrix, what always comes first: the number of columns or the number of rows? 5. What are the three types of elementary row operations ? Sample Answers (Note: for full credit, you should always either rewrite the question or write your answer in a complete sentence that rephrases the question.) Kelsoe Kismet Linear Algebra Topic 01 Previews 1. An example of an equation that is not linear is: x2 + 3x + 4 = 0. 2. A set of two linear equations can have either 0, 1, or infinitely many solutions. 3. A matrix is a rectangular grid of numbers. 4. The number of rows always comes first when specifying the size of a matrix. 5. The three types of elementary row operations are replacement, interchanging, and scaling. ...

#### FourPres

Toledo, CS 336
Excerpt: ... ng Ax = many-different-right-hand-sides first applies Gauss Elimination to A (and only does this once) to get the so-called LU factorization of A. And then, since A = LU, the algebraic problem becomes one of solving LUx = b. If we let y represent Ux, then we need to solve the two problems: Solve Ly = b for y and then Solve Ux = y for the unknown vector x. But! The coefficient matrices in these two problems are triangular, and hence the systems are easy to solve efficiently (using forward and backward substitution). Example: = Compute the LU factorization of the matrix A = ( 2 ( 4 ( -1 ( -2 4 -2 6 14 0 14 10 7 4 -1 0 2 ) ) ) ) We will apply Gauss Elimination to A in a very careful manner. The U matrix will be the result of the elimination process. The L matrix will be constructed from the row multipliers used in the elimination process. Eliminate entries below diagonal term A(i,i) for i = 1 Performing the following elementary row operations gives: R_2 <- R_2 (4/2) *R_1 R_3 <- R_3 - (-1/2)*R_1 ...

#### section1.3

UConn, MATH 1070
Excerpt: ... write an augmented matrix which is a rectangular array of the coefficients of the variables augmented with the constant terms. For example: x -3y +4z = -4 3x -7y +8z = -8 -4x +6y -z = 7 Goal: Use the elementary row operations to rewrite the augmented matrix so that it has 1's on the main diagonal and 0's below each of these 1's. 4 Elementary Row Operations : 1. Interchange the ith row and the jth row. Ri Rj 2. Multiply each member of the ith row by a non-zero constant k. kRi Ri 3. Replace each element in the ith row with the corresponding element of the ith row plus k times the jth row. Ri + kRj Ri Example 2: Solve the following system of equations using the augmented matrix method and Guass Elimination. x -3y +4z = -4 3x -7y +8z = -8 -4x +6y -z = 7 5 Example 2 continued 6 Example 3: Solve the following system of equations using the augmented matrix method and Guass Elimination. x +2y +z x +2y +2z y +z y -2z -u +2u -u +3u = -2 = 9 = -2 = 4 7 Example 3 continued 8 Example 4: Investments Jenni ...

#### section1.3

UConn, MATH 1070
Excerpt: ... we can write an augmented matrix which is a rectangular array of the coecients of the variables augmented with the constant terms. For example: x 3y +4z = 4 3x 7y +8z = 8 4x +6y z = 7 Goal: Use the elementary row operations to rewrite the augmented matrix so that it has 1s on the main diagonal and 0s below each of these 1s. 4 Elementary Row Operations : 1. Interchange the ith row and the jth row. Ri Rj 2. Multiply each member of the ith row by a non-zero constant k. kRi Ri 3. Replace each element in the ith row with the corresponding element of the ith row plus k times the jth row. Ri + kRj Ri Example 2: Solve the following system of equations using the augmented matrix method and Guass Elimination. x 3y +4z = 4 3x 7y +8z = 8 4x +6y z = 7 5 Example 2 continued 6 Example 3: Solve the following system of equations using the augmented matrix method and Guass Elimination. x +2y +z x +2y +2z y +z y 2z u +2u u +3u = 2 = 9 = 2 = 4 7 ...

#### l5

UCSD, MATH 102
Excerpt: ... 2x4 = 0 set to any value and we can solve for the other variables: . x3 + 2x4 = 0 2 2 0 1 This gives the general solution x = x4 + x2 . 2 0 1 0 The null space of A is therefore spanned by the two vectors above. Since these are not parallel it is not spanned by only one of them. A matrix is said to be in Row Echelon Form (step-like form) if Each leading entry (i.e. left most nonzero entry) of a row is in a column to the right of the leading entry of the row above it, e.g. 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 A matrix is said to be in Reduced Row Echelon Form if (i) It is in row echelon form, and (ii) Each leading non-zero entry is 1 and (iii) The leading entry in each row is the only non-zero entry in its column, e.g. 1 0 0 0 0 00 10 001 000 000 0 0 0 1 0 1 2 Recall the Elementary Row Operations on a matrix are 1. Add to one row a multiple o ...

#### lecture11

Michigan, IOE 510
Excerpt: ... i = l. At the last step, all information needed for the next iteration is saved. Storage: O(m), time: O(m3 + mn) NOTES IOE 510 Linear Programming I 65 Revised simplex idea Simplify computations by maintaining B 1 . When new basis, B, is created, compute B 1 using B 1 , rather then from scratch. Denition 3.4 Given a matrix (not necessarily square) the operation of adding a constant multiple of one row to the same or to another row is called an elementary row operation. Observation: elementary row operation on C is equivalent to forming QC, where Q is a suitable matrix. More precisely, to multiply row j by and add it to row i, set Q = I + Dij , where Dij is a matrix with in position i, j, and 0 elsewhere. Goal: Compute B 1 from B 1 by a sequence of at most m elementary row operations . Slide 65 NOTES IOE 510 Linear Programming I 66 Computing B 1 from B 1 B = AB(1) AB(m) and B = AB(1) AB(l1) Aj AB(l+1) AB(m) , so that ...

#### Week10tutes

Allan Hancock College, ICS 135
Excerpt: ... eck your answers by dierentiating. Ex. 2. Calculate the following denite integrals: 5 10 4 (a) 1 1 log t dt sin1 z dz 0 (b) 0 ze z dz (c) 0 tan1 d a (d) (e) 0 x sin x dx (f) 0 x2 ex dx, for a > 0 . Ex. 3. Calculate the following integrals: x d (a) cos(t2 + ) dt dx 0 x2 d (b) (3t3 t + 2)4 dt dx 1 x2 +3x d 2t (c) dt . 47 dx 2x t Ex. 4. For each of the following chemical reactions, determine the balanced chemical equation: (i) reactants: Al and O2 ; product Al2 O3 ; (ii) reactants: C2 H6 and O2 ; products CO2 and H2 O ; (iii) reactants: Pb O2 and HCl ; products Pb Cl2 , Cl2 and H2 O ; (iv) reactants: C2 H5 OH and O2 ; products CO2 and H2 O ; (v) reactants: Mn O2 , H2 SO4 and H2 C2 O4 ; products Mn SO4 , CO2 and H2 O . Ex. 5. By using suitable elementary row operations , show each of 2 3 7 1 3 1 2 3 4 2 3 7 1 5 1 2 3 4 ( )( )( )( ) 2 3 6 1 9 = 2 , det 1 ...