Homogeneous equations, Linear independence

Size: px
Start display at page:

Download "Homogeneous equations, Linear independence"

Transcription

1 Homogeneous equations, Linear independence 1. Homogeneous equations: Ex 1: Consider system: B" #B# œ! B" #B3 œ! B B œ! # $ Matrix equation: Ô " #! Ô B " Ô! "! # B # œ! œ 0Þ Ð3Ñ Õ! " " ØÕB Ø Õ! Ø $ Homogeneous equation: At least one solution: Ex œ 0. x œ 0Þ Other solutions called nontrivial solutions. Theorem 1: A nontrivial solution of Ð$Ñ exists iff [if and only if] the system has at least one free variable in row echelon form. The same is true for any homogeneous system of equations. Proof: If there are no free variables, there is only one solution and that must be the trivial solution. Conversely, if there are free variables, then they can be non-zero, and there is a nontrivial solution. Ex 2: Reduce the system above: Ô " #! l! Ô " #! l! as before "! # l! Ä! " " l! Õ! " " l! Ø Õ!!! l! Ø

2 Ê B #B œ!à B B œ!à!œ!þ " # # $ Note that B $ œ free variable (non-pivot); hence general solution is B# œ B$ à B" œ #B# œ #B$ Þ Ô B " Ô #B $ Ô # x œ B# œ B $ œ B$ " ß ÕB Ø Õ B Ø Õ " Ø $ $ B $ arbitrary: straight line - Parametric vector form of solution. Theorem 2: A homogeneous system always has a nontrivial solution if the number of equations is less than the number of unknowns. Pf: If we perform a Gaussian elimination on the system, then the reduced augmented matrix has the form:

3 Ô " + "# + "$ á l!!! " + #% á l!!!! " + $& á l ã á l ã Õ ã l! Ø with the remaining rows zeroes on the left side. If the number of equations is less than the number of unknowns, then not every column can have a 1 in it, so there are free variables. By previous theorem, there are nontrivial solutions.

4 1. Inhomogeneous equations: [we should briefly mention the relationship between homogeneous and inhomogeneous equations:] Consider general system: Ex œ bþ (1)(1) Suppose p is a particular solution of (1), so Ep œ b. If x is any other solution of (1), we still have Ex œ b. Subtracting the two equations: So v2 œ x p Ex Ep œ 0 Ê EÐx pñ œ 0Þ satisfies the homogeneous equation. Generally: Theorem 1: M0 p is a particular solution of (1), then for any other solution x, we have that v2 œ x p solves the homogeneous equation (i.e., with b œ 0). Thus every solution x of (1) can be written x œ p v 2, where v 2 is a solution of the homogeneous equation. 2. Application: Network flows Traffic pattern at Drummond Square:

5 Quantities in cars/min. What are the flows on the inside streets? One equation for each node: B B B œ %! " $ % B B œ #!! " # B B B œ "!! # $ & B B œ '! % & Ô "! " "! l %! " "!!! l #!!! " 1! " l "!! Õ!!! " " l '! Ø

6 Ô "! " "! l %!! " " "! l "'!! " 1! " l "!! Õ!!! " " l '! Ø Ô "! " "! l %!! " " "! l "'!! " 1! " l "!! Õ!!! " " l '! Ø Ô "! " "! l %!! " " "! l "'!!! 0 " " l '! Õ!!! " " l '! Ø Ô "! " "! l %!! " " "! l "'!!!! " " l '! Õ!!! " " l '! Ø Ô "! "! " l "!!! " "! " l "!!!!! " " l '! Õ!!!!! l! Ø So: B œ"!! B B " $ & B œ"!! B B # $ & B œ '! B ß % & where BßB $ & are free.

7 Constraint: if for example all flows have to be positive; then we require B 3! for all 3. Therefore: This corresponds to a region in the BßB! $ & "!! Ÿ B B Ÿ "!! $ & B Ÿ '! & BßB $ & plane - can be plotted if desired. If they closed off road B$ and Bß & then we have B$ œb& œ!, so that B" œ 100ß B# œ " 00, B% œ '! note that then traffic flow becomes uniquely determined. Definition 1: A collection of vectors v" ßv# ßáßv8 is linearly independent if no vector in the collection is a linear combination of the others. Equivalently, Definition 2: A collection of vectors v" ßáßv8 is linearly independent if the only way we can have - v - v á - v œ 0 is if all of the - œ!þ Equivalence of the definitions: Def 1 Ê Def 2 " " # # If no vector is a linear combination of the others, then if -" v" -# v# á -8v8 œ 0 we will show that -ßáß- " 8 have also to be 0. Proof: Suppose not (for contradiction). Without loss of generality, assume -" Á! (proof works same way otherwise). Then we have: v" œ -# Î-" v# á -8 Î-" v8ß

8 contradicting that no vector is a combination of the others. Thus the - 3 all have to be 0 as desired. Note: If W# is a collection of vectors and W" is a subcollection of W2ß then If W # is linearly independent Ê no vector in W # is a linear combination of the others Ê no vector in W" is a linear combination of the others (since every vector in W is also in W ) " # Ê W is linearly independent. " Logically equivalent [contrapositive] If W " is linearly dependent (i.e., not independent) Ê W is linearly dependent # [ These are stated more formally in the book as theorems.] Theorem 2: P/> W œ v" ßáß v8 be a collection of vectors in. Then W is linearly dependent if and only if one of the vectors v 3 is a linear combination of the previous ones v ßáßv Þ " 3 " Proof: ( Ê ) If W is linearly dependent, then there is a set of constnats - 3 not all! such that Let - 5 zero, and -" v" á -8v8 œ 0. be the last non-zero coefficient. Then the rest of the coefficients are -" v" -# v# á -5 " v5 " -5v5 œ 0 Ä v5 œ -" Î-5 v" -# Î-5 v# á -5 " Î-5v5 " i.e. one of the vectors is a linear combination of the previous ones. ( É ) Obvious..

9 3. Checking for linear independence: Example 2: Consider the vectors " " " v" œ ß v# œ ß v$ œ " "! Are they linearly independent? Ê -" v" -# v# -$ v$ œ œ! " # $ Reduced matrix: - - œ! " # " " " l! " "! l! Ä " " " l!! # " l! Ä " " " l!! " "Î# l! Conclude: there are free variables. By theorem on homogeneous equations there is a nontrivial solution, so - need not be!. Thus not all - 3 must be! Ê not linearly independent. [note that if number of vectors is greater than the size of the vectors, this will always happen]. More generally thus: 3

10 Theorem 3: In 8, if we have more than 8 vectors, they cannot be linearly independent. From above we have: Algorithm: To check whether vectors are linearly independent, form a matrix with them as columns, and row reduce. (a) If reduced matrix has free variables (i.e., b a non-pivot column), then they are not independent. (b) If there are no free variables (i.e., there are no nonpivot columns), they are independent.

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above: Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A

More information

1.5 SOLUTION SETS OF LINEAR SYSTEMS

1.5 SOLUTION SETS OF LINEAR SYSTEMS 1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d. DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

More information

1.2 Solving a System of Linear Equations

1.2 Solving a System of Linear Equations 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

More information

Bud row 1. Chips row 2. Coors. Bud. row 3 Milk. Chips. Cheesies. Coors row 4 Cheesies. Diapers. Milk. Diapers

Bud row 1. Chips row 2. Coors. Bud. row 3 Milk. Chips. Cheesies. Coors row 4 Cheesies. Diapers. Milk. Diapers Ð ØÖ ØÝ ÜØ ÖÒ Ð Ë Ñ Ð Ö ØÝ Ó Ø ÓÖ Ð ØØÖ ÙØ Ö ØÓÔ Ö Êº È ÐÑ Ö ½ Ò Ö ØÓ ÐÓÙØ Ó ¾ ¾ ½ Î Ú ÑÓ ÁÒº ¾ ÛÓÓ ÐÚ È ØØ ÙÖ È Ô ÐÑ ÖÚ Ú ÑÓºÓÑ ÓÑÔÙØ Ö Ë Ò Ô ÖØÑ ÒØ ÖÒ Å ÐÐÓÒ ÍÒ Ú Ö ØÝ ¼¼¼ ÓÖ Ú È ØØ ÙÖ È Ö ØÓ ºÑÙº Ù

More information

Section 1.7 22 Continued

Section 1.7 22 Continued Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

Controllability and Observability

Controllability and Observability Controllability and Observability Controllability and observability represent two major concepts of modern control system theory These concepts were introduced by R Kalman in 1960 They can be roughly defined

More information

Section 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)

Section 8.2 Solving a System of Equations Using Matrices (Guassian Elimination) Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

5 Homogeneous systems

5 Homogeneous systems 5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

ÔØÖ ÄÒÖ Ç ÐÐØÓÖ ß ÇÒ Ö Ó ÖÓÑ º½ ÇÚÖÚÛ ÏØ Ó Ø ØÒ Ú Ò ÓÑÑÓÒ ÔÖÓÔØÓÒ Ó Ñ ÛÚ ÒÖØ Ý Öع ÕÙ ÖÑÓØ ØØÓÒ Ó ÓÑÔÐÜ ÑÓÐÙÐ Ú ÒÖÖ ÔØÖ Ø ÐØÖ Ò ÑÒØ Ð Ò ÑÖÓÛÚ ÚØÝ Ò ÖÒØÖ ÐÓ Ì ÙÑÐ ÑÔÐ ÖÑÓÒ Ó Ð¹ ÐØÓÖ ÔÐÝ ØÖÖÒ ÖÓÐ Ò Ø ÙÒÖ

More information

Æ ÒØ Ò Ö Ø ÓÒ Ó ÊÓØ Ø Ò ÏÓÖ ÓÖ Ë ÙÐ ÆÝ Ö Ø ÅÙ Ð ÂÓ ÒÒ Đ ÖØÒ Ö Ò ÏÓÐ Ò ËÐ ÒÝ ØÖ Øº Ò Ö Ø Ò ¹ÕÙ Ð ØÝ ÙÐ ÓÖ ÖÓØ Ø Ò ÛÓÖ ÓÖ Ö Ø Ð Ø Ò ÐÐ ØÙ Ø ÓÒ Û Ö ÖØ Ò Ø ÆÒ Ð Ú Ð ÑÙ Ø Ù Ö¹ ÒØ Ù Ò Ò Ù ØÖ Ð ÔÐ ÒØ Ó Ô Ø Ð

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

Ä ØÙÖ ËÐ ÁÒÚ ØÑ ÒØ Ò ÐÝ ½ ÌÖ Ò Ò ÁÒØÖÓ ØÓ ÌË Ó Ð ØÖ Ò Ø ÖÑ ÒÓÐÓ Ý ÜÔÐ Ò Ä Û Ó ÇÒ ÈÖ Ò Ö ØÖ ÐÙÐ Ø Ö ÔÐ Ø Ò ÔÓÖØ ÓÐ Ó Ó ÓÒ ÜÔÐ Ò Ö Ð Ø ÓÒ Ô Ö ØÖ Ò Ê ÔÐ Ø ÓÒ ËÔÓØ Ê Ø ÓÖÛ Ö Ê Ø Ä ØÙÖ ËÐ ÁÒÚ ØÑ ÒØ Ò ÐÝ ¾ ÇÖ

More information

Ê ÔÓÒ Ú Ì ÒÛ Ö Î Ù Ð Þ Ø ÓÒ Ó Ä Ö Ó Ö Ô Ø Ø Ý Ã ÒÒ Ø Ò ÖØ Ø ÓÒ Ù Ñ ØØ Ò Ô ÖØ Ð ÙÐ ÐÐÑ ÒØ Ó Ø Ö ÕÙ Ö Ñ ÒØ ÓÖ Ø Ö Ó ÓØÓÖ Ó È ÐÓ ÓÔ Ý Ô ÖØÑ ÒØ Ó ÓÑÔÙØ Ö Ë Ò Æ Û ÓÖ ÍÒ Ú Ö ØÝ Ë ÔØ Ñ Ö ¾¼¼¾ ÔÔÖÓÚ Ô Ã ÒÒ Ø Ò

More information

ÔØ Ö ½ ÊÇÍÌÁÆ ÁÆ ÅÇ ÁÄ ÀÇ Æ ÌÏÇÊÃË Å Ãº Å Ö Ò Ò Ë Ñ Ö Êº Ô ÖØÑ ÒØ Ó ÓÑÔÙØ Ö Ë Ò ËØ Ø ÍÒ Ú Ö ØÝ Ó Æ Û ÓÖ Ø ËØÓÒÝ ÖÓÓ ËØÓÒÝ ÖÓÓ Æ ½½ ¹ ¼¼ ØÖ Ø Æ ÒØ ÝÒ Ñ ÖÓÙØ Ò ÓÒ Ó Ø Ý ÐÐ Ò Ò ÑÓ Ð Ó Ò ØÛÓÖ º ÁÒ Ø Ö ÒØ Ô

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

ÆÏ ÈÈÊÇÀ ÌÇ Ëµ ÁÆÎÆÌÇÊ ËËÌÅË ÂÒ¹ÉÒ ÀÙ ÅÒÙØÙÖÒ ÒÒÖÒ Ó ØÓÒ ÍÒÚÖ ØÝ ËÓÖ ÆÒÒÙÙÐ Ý Ò Ï¹Ó ÓÒ Ý ÐØÖÐ Ò ÓÑÔÙØÖ ÒÒÖÒ ÍÒÚÖ ØÝ Ó Å Ù ØØ ÑÖ Ø ÖÙÖÝ ØÖØ ÁÒ Ø ÔÔÖ Û ÓÒ Ö ÔÖÓ ÖÚÛ Ëµ ÒÚÒØÓÖÝ Ý ØÑ ÛØ ÒÔÒÒØ Ò ÒØÐÐÝ ØÖÙØ

More information

ÁÒÖÒ ÓÖ Ó ÖÚØÓÒ Ó ÒØÖØ «Ù ÓÒ ÔÖÓ º ËÙ ÒÒ ØÐÚ Ò ÔÖØÑÒØ Ó Ó ØØ Ø ÅÐ ËÖÒ Ò ÔÖØÑÒØ Ó ËØØ Ø Ò ÇÔÖØÓÒ Ê Ö ÍÒÚÖ ØÝ Ó ÓÔÒÒ ÒÑÖ ØÖØ ØÑØÓÒ Ó ÔÖÑØÖ Ò «Ù ÓÒ ÑÓÐ Ù ÙÐÐÝ ÓÒ Ó Ö¹ ÚØÓÒ Ó Ø ÔÖÓ Ø ÖØ ØÑ ÔÓÒØ º ÀÖ Û ÒÚ ØØ

More information

Ø Ö ØÒ ÓÑÔ Ð Â Ú ÈÖÓ º ÓÒÒ Ø ÔÖÓÚ º Ø Þº µ ÔÖ Ð ¾ ¾¼¼½ ØÖ Ø ÓÖ ÕÙ Ø ÓÑ Ø Ñ ÒÓÛ Ñ Ö Ó Â Ú Î ÖØÙ Ð Å Ò ÂÎÅ µ ÓÒ Â٠عÁÒ¹Ì Ñ ÂÁ̵ Ò ¹Ç ¹Ì Ñ Ç̵ ÓÑÔ Ð Ö Û Ø Óҹع Ý ÓÔØ Ñ Þ Ø ÓÒ Ú Ò ÙÒØ Ò Ø Ö ÈÖÓ ÙØ ÖÙÒÒ Ò

More information

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

More information

Row Echelon Form and Reduced Row Echelon Form

Row Echelon Form and Reduced Row Echelon Form These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

Solving Linear Systems, Continued and The Inverse of a Matrix

Solving Linear Systems, Continued and The Inverse of a Matrix , Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

Primitives. Ad Hoc Network. (a) User Applications Distributed Primitives. Routing Protocol. Ad Hoc Network. (b)

Primitives. Ad Hoc Network. (a) User Applications Distributed Primitives. Routing Protocol. Ad Hoc Network. (b) Ï Ö Ð Æ ØÛÓÖ ¼ ¾¼¼½µ ß ½ ÅÙØÙ Ð ÜÐÙ ÓÒ Ð ÓÖ Ø Ñ ÓÖ ÀÓ ÅÓ Ð Æ ØÛÓÖ Â ÒÒ Ö º Ï ÐØ Ö Â ÒÒ Ö Äº Ï Ð Æ Ø Ò Àº Î Ý Ô ÖØÑ ÒØ Ó ÓÑÔÙØ Ö Ë Ò Ì Ü ²Å ÍÒ Ú Ö ØÝ ÓÐÐ ËØ Ø ÓÒ Ì ¹ ½½¾ ¹Ñ Ð ÒÒÝÛ ºØ ÑÙº Ù Û Ð ºØ ÑÙº Ù

More information

Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 Spanning and Independence Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems 1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

More information

The CMS Silicon Strip Tracker and its Electronic Readout

The CMS Silicon Strip Tracker and its Electronic Readout The CMS Silicon Strip Tracker and its Electronic Readout Markus Friedl Dissertation May 2001 ÖØ Ø ÓÒ Ì ÅË Ë Ð ÓÒ ËØÖ Ô ÌÖ Ö Ò Ø Ð ØÖÓÒ Ê ÓÙØ ÔÖ ÒØ Ò Ô ÖØ Ð ÙÐ ÐÐÑ ÒØ Ó Ø Ö ÕÙ Ö Ñ ÒØ ÓÖ Ø Ö ÓØÓÖ Ó Ì Ò Ð

More information

(a) Hidden Terminal Problem. (b) Direct Interference. (c) Self Interference

(a) Hidden Terminal Problem. (b) Direct Interference. (c) Self Interference ØÖ ÙØ ÝÒ Ñ ÒÒ Ð Ë ÙÐ Ò ÓÖ ÀÓ Æ ØÛÓÖ ½ Ä ÙÒ Ó Ò ÂºÂº Ö ¹ÄÙÒ ¹ Ú Ë ÓÓÐ Ó Ò Ò Ö Ò ÍÒ Ú Ö ØÝ Ó Ð ÓÖÒ Ë ÒØ ÖÙÞ ¼ ¹Ñ Ð ÓÐ Ó ºÙ º Ù Î Ö ÓÒ ¼»½»¾¼¼¾ Ì Ö ØÝÔ Ó ÓÐÐ ÓÒ¹ Ö ÒÒ Ð ÔÖÓØÓÓÐ ÓÖ Ó Ò ØÛÓÖ Ö ÔÖ ÒØ º Ì ÔÖÓØÓÓÐ

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

ÔÔÖ Ò ÂÓÙÖÒÐ Ó ÓÑÔÙØÖ Ò ËÝ ØÑ ËÒ ÎÓк ½ ÆÓº ¾¼¼¼ ÔÔº ¾ß º ÈÖÐÑÒÖÝ ÚÖ ÓÒ Û Ò ÚÒ Ò ÖÝÔØÓÐÓÝ ß ÖÝÔØÓ ÈÖÓÒ ÄØÙÖ ÆÓØ Ò ÓÑÔÙØÖ ËÒ ÎÓк º ÑØ º ËÔÖÒÖ¹ÎÖÐ ½º Ì ËÙÖØÝ Ó Ø ÔÖ ÐÓ ÒÒ Å ÙØÒØØÓÒ Ó ÅÖ ÐÐÖ ÂÓ ÃÐÒ Ý ÈÐÐÔ

More information

ÓÒØÖÓÐ ËÝ Ø Ñ Ò Ò Ö Ò ÖÓÙÔ Ò ÙØÓÑ Ø ÓÒ Ì ÒÓÐÓ Ý ÖÓÙÔ Ö¹ÁÒ Ò Ö ÂÓ Ñ ÔÙØݵ Ø ØÖ ½ ¼ ¼ À Ò È ÓÒ ¼¾ ½¹ ¹½½¼¼ Ü ¼¾ ½¹ ¹ ¹Å Ð Ò Ö Ó Ñ ÖÒÙÒ ¹ Ò Ñ Ø «È ÓÒ ÓÒØÖÓÐ ËÝ Ø Ñ Ò Ò Ö Ò ÖÓÙÔ ÔйÁÒ Ò Ö Ó«Ö¹ÁÒ ÍÐÖ ÓÖ ÓÐØ

More information

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which Homogeneous systems of algebraic equations A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x + + a n x n = a

More information

ÐÓÒ¹Ü Ö ËÔÖ ½ ÖÖÐÐ ÙÆ Ò ÂÙÒ ÄÙ ËÒÓÖ ÍÒÚÖ Ý ÙÖÖÒ ÎÖ ÓÒ ÑÖ ¾ ½ Ö Ï ÙÝ ÖÑ ÖÙÙÖ Ó ÝÐ ÔÖ ÛÒ ÓÒ¹Ö Ò Ü¹ Ö ÒÓ Ó Ñ Ö ÕÙÐÝ Ò ÑÙÖݺ ÐÓÒ¹ Ü ÔÖ Ö ÓÖÐÐÝ ÖÖÞ Ò ÓÑ ÔÖÐ Ò ÕÙÒ Ò ÑÔÐ ÑÓÐ Ò ÖÑ Ó ÑÙÖÝ Ö ÕÙÐÝ ÝÐ ÚÓÐÐÝ Ýй ÔÖ

More information

ÉÙ ÖÝ Ò Ë Ñ ØÖÙØÙÖ Ø ÇÒ Ë Ñ Å Ø Ò Á Ë Ë Ê Ì Ì Á Ç Æ ÞÙÖ ÖÐ Ò ÙÒ Ñ Ò Ö ÓØÓÖ Ö ÖÙÑ Ò ØÙÖ Ð ÙÑ Öº Ö Öº Ò Øºµ Ñ ÁÒ ÓÖÑ Ø Ò Ö Ø Ò Ö Å Ø Ñ Ø ¹Æ ØÙÖÛ Ò ØÐ Ò ÙÐØĐ Ø ÁÁ ÀÙÑ ÓРعÍÒ Ú Ö ØĐ Ø ÞÙ ÖÐ Ò ÚÓÒ À ÖÖ Ôк¹ÁÒ

More information

é é ä ä é ö é é ò é ó é Ü ä Ü ä ä

é é ä ä é ö é é ò é ó é Ü ä Ü ä ä é é é ä èé èé ö ß é éé é é é ß ß ßß ß é é é é ä ä ä ä ä é ä ä éé é ä é é ä ä é ä ö é é ò é é ó é Üä Üää à ò éè ì é è èè è ó üü èè è ü è è é é ä éé óé ä ìò ì é ä é ó ó é é ó é éé é é Ü é é ò ò ò ä ää é

More information

MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

ÅÁÌ ½ º ÌÓÔ Ò Ì Ë ÁÒØ ÖÒ Ø Ê Ö ÈÖÓ Ð Ñ ËÔÖ Ò ¾¼¼¾ Ä ØÙÖ ½ ÖÙ ÖÝ ¾¼¼¾ Ä ØÙÖ Ö ÌÓÑ Ä ØÓÒ ËÖ ÇÑ Ö Ø ÑÓÒ Ï Ð ½º½ ÁÒØÖÓ ÙØ ÓÒ Ì Ð Û ÐÐ Ù Ú Ö Ð Ö Ö ÔÖÓ Ð Ñ Ø Ø Ö Ö Ð Ø ØÓ Ø ÁÒØ ÖÒ Øº Ð ØÙÖ Û ÐÐ Ù ÀÓÛ Ô ÖØ ÙÐ

More information

ÅÓÖ Ð À Þ Ö ÁÒ ÙÖ Ò Ò ËÓÑ ÓÐÐÙ ÓÒ ÁÒ Ð Ð Ö Ò Ò ¹ØÓ Ð ÖØ Å Ö Ø Ú Ö ÓÒ Ù Ù Ø ½ Ì Ú Ö ÓÒ ÖÙ ÖÝ ¾¼¼½ ØÖ Ø Ï ÓÒ Ö ÑÓ Ð Ó Ò ÙÖ Ò Ò ÓÐÐÙ ÓÒº Æ ÒØ Ö Ö Ò Ö ÕÙ Ö Ø ÓÒ ÙÑ Ö ØÓ Ø ÑÓÒ Ø ÖÝ ÓÑÔ Ò Ø ÓÒ Ò Ó ÐÓ º ÙØ Ø

More information

ÕÙ ØÝ ÌÖ Ò Ý ÁÒ Ø ØÙØ ÓÒ Ð ÁÒÚ ØÓÖ ÌÓ ÖÓ ÓÖ ÆÓØ ØÓ ÖÓ Ì Ó Ø ÆÓÖÛ Ò È ØÖÓÐ ÙÑ ÙÒ º Ê Ò Æ ÆÓÖ Ò ÖÒØ ÖÒ Ö ÆÓÖ Ò Ò ÆÓÖÛ Ò Ë ÓÓÐ Ó Å Ò Ñ ÒØ ½ Â ÒÙ ÖÝ ¾¼¼¼ ØÖ Ø Ì Ó Ø ØÓ Ò Ø ØÙØ ÓÒ Ð ÒÚ ØÓÖ Ó ØÖ Ò ÕÙ ØÝ Ö Ó

More information

Linear Algebra Notes

Linear Algebra Notes Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

More information

Ì ÍÆÁÎ ÊËÁÌ ÌÁË ÆÁ ˵ Ë Öº Ð º Ò Ö º ÚÓк ½ ÆÓº ½ ÔÖ Ð ¾¼¼¾ ½ ¹ ½ ÐÓ Ò Ò ÅÙÐØ ¹ ÀÞ ÒÚ ÖÓÒÑ ÒØ ÎÓ Ò º Ç ÐÓ Þ ÁÒÚ Ø È Ô Ö ØÖ Ø Ò ÓÚ ÖÚ Û Ó ÐÓ Ò Ò Ò Ó ÐÓ ØÓÖ Ð Ñ ÒØ ÔÖ ÒØ º ËÝ Ø Ñ Ø Ò Ó Ô¹ ÓÔ ÜÔÐ Ò Û ÐÐ Ø

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

ÔØ Ö Ê Ö ÓÐÓ Ý ÁÒ Ø ÔØ Ö Ø Ö Ñ Ò ÛÓÖ Ø Ø ÓÒ Ú ÐÓÔ ÔÖ ÒØ º Ì ÛÓÖ Ø ¹ Ø ÓÒ ÓÑÔÙØ Ö Ø ÒÓ Ø ÑÓ ÙÐ Û Ö Ø ÓÖÓÒ ÖÝ ØÖ ÑÓ Ð ÐÐ ÔÐ Ý Ò ÑÔÓÖØ ÒØ ÖÓÐ Û Ò Ó Ò ÙØÓÑ Ø Ú Ð Ò ÐÝ Û Ø ÓÖÓÒ ÖÝ Ò Ó Ö ¹ Ô Ý Ñ º Ì ÔØ Ö Ò Û

More information

3.1 State Space Models

3.1 State Space Models 31 State Space Models In this section we study state space models of continuous-time linear systems The corresponding results for discrete-time systems, obtained via duality with the continuous-time models,

More information

ÁÆÎÆÌÇÊ ÇÆÌÊÇÄ ÍÆÊÌÁÆ ÅƵ ÑÒ ÙÒÖØÒ Ø ÊÒ ÎÖÒ Ó ÊÒ Ø Á ¼ ÈÊÇÍÌÁÇÆ ÈÄÆÆÁÆ Æ ÇÆÌÊÇÄ ÁÒÚÒØÓÖÝ ÓÒØÖÓÐ ÙÒÖØÒ ÑÒµ ÁÒØÖÓÙØÓÒ ÊÒÓÑ ÚÖØÓÒ ÑÔÓÖØÒØ ÔÖØÐ ÚÖØÓÒ ÈÖÓÐÑ ØÖÙØÙÖ ÑÔÐ ØÓ ÖÔÖ ÒØ ÖÒÓÑÒ Ò Ø ÑÓÐ Ê Æ ÍÒÚÖ ØÝ Ø

More information

Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

More information

Universitat Autònoma de Barcelona

Universitat Autònoma de Barcelona Universitat Autònoma de Barcelona ÙÐØ Ø Ò Ë Ó ³ Ò ÒÝ Ö ÁÒ ÓÖÑ Ø ÇÒ Ø Ò Ò ÓÒ ØÖÙØ ÓÒ Ó ÒØ¹Ñ Ø ÁÒ Ø ØÙØ ÓÒ Å Ñ ÓÖ ÔÖ ÒØ Ô Ö Ò ÂÙ Ò ÒØÓÒ Ó ÊÓ Ö Ù Þ Ù Ð Ö Ô Ö ÓÔØ Ö Ð Ö Ù ÓØÓÖ Ò ÒÝ Ö Ò ÁÒ ÓÖÑ Ø ÐÐ Ø ÖÖ Å ¾¼¼½

More information

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

More information

PROTOCOLS FOR SECURE REMOTE DATABASE ACCESS WITH APPROXIMATE MATCHING

PROTOCOLS FOR SECURE REMOTE DATABASE ACCESS WITH APPROXIMATE MATCHING CERIAS Tech Report 2001-02 PROTOCOLS FOR SECURE REMOTE DATABASE ACCESS WITH APPROXIMATE MATCHING Wenliang Du, Mikhail J. Atallah Center for Education and Research in Information Assurance and Security

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

HTML Codes - Characters and symbols

HTML Codes - Characters and symbols ASCII Codes HTML Codes Conversion References Control Characters English version Versión español Click here to add this link to your favorites. HTML Codes - Characters and symbols Standard ASCII set, HTML

More information

ÍÒ Ö Ø Ò Ò Ø ÒØ ÖÔÖ ÁÒ ÓÖÑ Ø ÓÒ ËÝ Ø Ñ Ì ÒÓÐÓ Ý Ó Ò ÊÈ Ö Ï Ò Ö ØØÔ»»ÛÛÛº Ò º Ù¹ ÖÐ Òº» Û Ò Ö ÁÒ Ø ØÙØ ĐÙÖ ÁÒ ÓÖÑ Ø Ö ÍÒ Ú Ö ØĐ Ø ÖÐ Ò Ì Ù ØÖº ¹½ ½ ÖÐ Ò ÖÑ ÒÝ ¹Ñ Ð Û Ò º Ù¹ ÖÐ Òº ÔÖ Ð ¾ ¾¼¼¼ ÌÙØÓÖ Ð Ø Ø

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

More information

The ASCII Character Set

The ASCII Character Set The ASCII Character Set The American Standard Code for Information Interchange or ASCII assigns values between 0 and 255 for upper and lower case letters, numeric digits, punctuation marks and other symbols.

More information

>?@CHJLC?JHLL >?@CHJLC?JHLL \ >?@CH >?JHLL >?JLC?!"#$%!&'!()+,-.+/0-1,.20.3456,.13.71+1,89:0-;,.-

More information

Linear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

Linear Equations ! 25 30 35$ &  350 150% &  11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

ØÙÖ Ò Ö Ø ÓÒ Ý ÁÑ Ø Ø ÓÒ ÖÓÑ ÀÙÑ Ò Ú ÓÖ ØÓ ÓÑÔÙØ Ö Ö Ø Ö Ò Ñ Ø ÓÒ ÖØ Ø ÓÒ ÞÙÖ ÖÐ Ò ÙÒ Ö Ó ØÓÖ Ö ÁÒ Ò ÙÖÛ Ò Ø Ò Ö Æ ØÙÖÛ Ò ØÐ ¹Ì Ò Ò ÙÐØĐ Ø Ò Ö ÍÒ Ú Ö ØĐ Ø Ë ÖÐ Ò ÚÓÖ Ð Ø ÚÓÒ Å Ð Ã ÔÔ Ë Ö ÖĐÙ Ò ¾¼¼ Ò ÎÓÖ

More information

Examination paper for TMA4115 Matematikk 3

Examination paper for TMA4115 Matematikk 3 Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99

More information

Å Ò Ñ ÒØ Ö Ø ØÙÖ Ö Ñ ÛÓÖ ÓÖ Ø Ú Æ ØÛÓÖ Ð ÒϺ ÓÒ Â Ñ Èº ºËØ Ö ÒÞ Ñ Ñ Ò Ð Ü Ò ÖκÃÓÒ Ø ÒØ ÒÓÙ ÆÌ ÒÓÐÓ Î Ö ÞÓÒ ÓÐÙÑ ÍÒ Ú Ö ØÝÝ ¼ÂÙÒ ¾¼¼ ÝØ ÊÄÙÒ ÖÓÒØÖ Ø ¼ ¼¾¹ ¹ ¹¼½ ½º Ì ÛÓÖ Û ÔÓÒ ÓÖ ÝØ Ò Ú Ò Ê Ö ÈÖÓ Ø ÒÝ

More information

ÆÓØ Ä ØÙÖ Ð Ñ Ø ØÖÙ ÙØ ÓÒ ØÓ Á ¼ ØÙ ÒØ ÓÖ ÐÐ ÓØ Ö Ö Ø Ö ÖÚ Á ¼ ÈÊÇ Í ÌÁÇÆ ÈÄ ÆÆÁÆ Æ ÇÆÌÊÇÄ Ê Æ Ô ÖØÑ ÒØ Ó ÁÒ Ù ØÖ Ð Ò Ò Ö Ò ÍÒ Ú Ö ØÝ Ø Ù«ÐÓ ¹ ËØ Ø ÍÒ Ú Ö ØÝ Ó Æ Û ÓÖ Ò Ù«ÐÓº Ù Á ¼ ÈÊÇ Í ÌÁÇÆ ÈÄ ÆÆÁÆ Æ

More information

ÀÖÖÐ ÈÐÑÒØ Ò ÆØÛÓÖ Ò ÈÖÓÐÑ ËÙÔØÓ Ù Ñ ÅÝÖ ÓÒ Ý ÃÑ ÅÙÒÐ Þ ÅÝ ½ ¾¼¼¼ ØÖØ ÁÒ Ø ÔÔÖ Û Ú Ø Ö Ø ÓÒ ØÒعÔÔÖÓÜÑØÓÒ ÓÖ ÒÙÑÖ Ó ÐÝÖ ÒØÛÓÖ Ò ÔÖÓÐÑ º Ï Ò Ý ÑÓÐÒ ÖÖÐ Ò ÛÖ Ö ÔÐ Ò ÐÝÖ Ò ÐÝÖ Ø Ü ÔÖÒØ Ó Ø ÑÒ ÓÙÒ Ñ ÖØ µº

More information

ÓÑÔ Ö Ø Ú Ê Ú Û Ó ÊÓ ÓØ ÈÖÓ Ö ÑÑ Ò Ä Ò Ù ÁÞÞ Ø È Ñ Ö ÓÖÝ À Ö Ù Ù Ø ½ ¾¼¼½ ØÖ Ø ÁÒ Ø Ô Ô Ö Û Ñ ÓÑÔ Ö Ø Ú Ö Ú Û Ó Ú Ö ØÝ Ó ÒØ ÖÑ Ø ¹Ð Ú Ð ÖÓ ÓØ Ð Ò Ù Ø Ø Ú Ñ Ö Ò Ö ÒØ Ý Ö º Ï Ð Ó Ö ÖÓ ÓØ ÔÖÓ Ö ÑÑ Ò Ð Ò Ù

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Author manuscript, published in "1st International IBM Cloud Academy Conference - ICA CON 2012 (2012)" hal-00684866, version 1-20 Apr 2012

Author manuscript, published in 1st International IBM Cloud Academy Conference - ICA CON 2012 (2012) hal-00684866, version 1-20 Apr 2012 Author manuscript, published in "1st International IBM Cloud Academy Conference - ICA CON 2012 (2012)" Á ÇÆ ¾¼½¾ ÌÓÛ Ö Ë Ð Ð Ø Å Ò Ñ ÒØ ÓÖ Å Ô¹Ê Ù ¹ Ø ¹ÁÒØ Ò Ú ÔÔÐ Ø ÓÒ ÓÒ ÐÓÙ Ò ÀÝ Ö ÁÒ Ö ØÖÙØÙÖ Ö Ð ÒØÓÒ

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

FRAME. ... Data Slot S. Data Slot 1 Data Slot 2 C T S R T S. No. of Simultaneous Users. User 1 User 2 User 3. User U. No.

FRAME. ... Data Slot S. Data Slot 1 Data Slot 2 C T S R T S. No. of Simultaneous Users. User 1 User 2 User 3. User U. No. ÂÓÙÖÒ ÐÓ ÁÒØ ÖÓÒÒ Ø ÓÒÆ ØÛÓÖ ÎÓк¾ ÆÓº½ ¾¼¼½µ ¹ ÏÓÖÐ Ë ÒØ ÈÙ Ð Ò ÓÑÔ ÒÝ È Ê ÇÊÅ Æ Î ÄÍ ÌÁÇÆÇ Ê ÉÍ ËÌ¹Ì Å» Å ÈÊÇÌÇ ÇÄ ÇÊÏÁÊ Ä ËËÆ ÌÏÇÊÃË ÒØ Ö ÓÖÊ Ö ÒÏ Ö Ð ÅÓ Ð ØÝ Ò Æ ØÛÓÖ Ò Ê ÏŠƵ Ô ÖØÑ ÒØÓ ÓÑÔÙØ ÖË Ò

More information

ÌÖ Ò Ø ÓÒ¹ Ö Ò Ò ÅÒ ÑÓ ÝÒ È Ö¹ØÓ¹È Ö ËØ ÒÓ Ö Ô ËØÓÖ ËÝ Ø Ñ Ì ÑÓØ Ý ÊÓ Ó ½ Ò ËØ Ú Ò À Ò ¾ ¾ ½ ËÔÖ ÒØ Ú Ò Ì ÒÓÐÓ Ý Ä ÓÖ ØÓÖÝ ÙÖÐ Ò Ñ ¼½¼ ÍË ÍÒ Ú Ö ØÝ Ó Ñ Ö ÓÑÔÙØ Ö Ä ÓÖ ØÓÖÝ Ñ Ö ¼ Íà ØÖ Øº ÅÒ ÑÓ ÝÒ Ô Ö¹ØÓ¹Ô

More information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

More information

(a) Original Images. (b) Stitched Image

(a) Original Images. (b) Stitched Image ÁÅ Ê ÁËÌÊ ÌÁÇÆ ÁÆ ÌÀ Å ÌÄ ÆÎÁÊÇÆÅ ÆÌ Åº ÅÙ ÖÓÚ º ÈÖÓ Þ ÁÒ Ø ØÙØ Ó Ñ Ð Ì ÒÓÐÓ Ý Ô ÖØÑ ÒØ Ó ÓÑÔÙØ Ò Ò ÓÒØÖÓÐ Ò Ò Ö Ò ØÖ Ø Ì Ô Ô Ö ÚÓØ ØÓ ÔÓ Ð Ø Ó ÓÑ ØÖ ÑÓ Ø ÓÒ Ó Ñ ØÓ Ò Ð ÓÒÒ Ø ÓÒ Ó Ô Ö Ø Ò Ó ÓÚ ÖÐ Ý Ò Ñ

More information

Lecture Notes 2: Matrices as Systems of Linear Equations

Lecture Notes 2: Matrices as Systems of Linear Equations 2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

More information

Open Programmable Architecture for Java-enabled Network Devices

Open Programmable Architecture for Java-enabled Network Devices Open Programmable Architecture for Java-enabled Network Devices Tal Lavian, Nortel Networks - Advanced Technology Center Robert F. Jaeger, University of Maryland Jeffrey K. Hollingsworth, University of

More information

ÇÔ Ò ÈÖÓ Ð Ñ Ò Ø ¹Ë Ö Ò È Ö¹ØÓ¹È Ö ËÝ Ø Ñ Æ Ð Û Ò À ØÓÖ Ö ¹ÅÓÐ Ò Ò Ú ÖÐÝ Ò ËØ Ò ÓÖ ÍÒ Ú Ö ØÝ ËØ Ò ÓÖ ¼ ÍË Û Ò ØÓÖ Ý Ò º Ø Ò ÓÖ º Ù ØØÔ»»ÛÛÛ¹ º Ø Ò ÓÖ º Ù ØÖ Øº ÁÒ È Ö¹ÌÓ¹È Ö È¾Èµ Ý Ø Ñ ÙØÓÒÓÑÓÙ ÓÑÔÙØ Ö

More information

Sliding Window ... Basic Window S[0] S[k 1] S[k] Digests Digests Digests

Sliding Window ... Basic Window S[0] S[k 1] S[k] Digests Digests Digests ËØØËØÖÑ ËØØ ØÐ ÅÓØÓÖ Ó ÌÓÙ Ó Ø ËØÖÑ ÊÐ ÌÑ ÙÝÙ Ù Ë ÓÙÖØ Á ØØÙØ Ó ÅØÑØÐ Ë ÔÖØÑØ Ó ÓÑÔÙØÖ Ë ÆÛ ÓÖ ÍÚÖ ØÝ ÝÙÝÙ ºÝÙºÙ ØÖØ Ó Ö Ø ÔÖÓÐÑ Ó ÑÓØÓÖ Ø Ó ØÓÙ Ó ØÑ Ö Ø ØÖÑ ÓÐ Ó Ñ Ó Ó ØѺ Á ØÓ ØÓ Ð ØÖÑ ØØ ¹ Ø Ù ÚÖ ØÖ

More information

Client URL. List of object servers that contain object

Client URL. List of object servers that contain object ÄÓ Ø Ò ÓÔ Ó Ç Ø Í Ò Ø ÓÑ Ò Æ Ñ ËÝ Ø Ñ ÂÙ Ã Ò Ö Ù Ã Ø Ïº ÊÓ ÁÒ Ø ØÙØ ÙÖ ÓÑ ËÓÔ ÒØ ÔÓÐ Ö Ò Ò ÖÓ ÙÖ ÓѺ Ö Â Ñ Ïº ÊÓ ÖØ Ö Ò Ì Ð ÓÑ ß Æ Ì Á Ý Ð ÅÓÙÐ Ò ÙÜ Ö Ò ØÖ Ø ½ ÁÒØÖÓ ÙØ ÓÒ ÁÒ ÓÖ Ö ØÓ Ö Ù Ú Ö Ð Ý Ò Ò ¹

More information

Ë ÓÒ Ð ØÝ Ò Ö ÙÐØÙÖ Ð ÓÑÑÓ ØÝ ÙØÙÖ Ö Ø Ò Ë Ö Ò Ò Ô ÖØÑ ÒØ Ó Ò Ò ÓÔ Ò Ò Ù Ò Ë ÓÓÐ ÊÓ Ò ÖÒ ÐÐ ½ ù½ ¼ Ö Ö Ö ÒÑ Ö Ì Ä ½ ½ ½ ¼¼ ¹Ñ Ð Óº º Ñ Ö ½ Ì ÙØ ÓÖ Ø Ò ÓÖ ÐÔ ÙÐ Ø Ò ÖÓÑ Â Ô Ö ĐÙÐÓÛ Ò ÓÑÑ ÒØ Ò Ù Ø ÓÒ ÖÓÑ

More information

How to create OpenDocument URL s with SAP BusinessObjects BI 4.0

How to create OpenDocument URL s with SAP BusinessObjects BI 4.0 How to create OpenDocument URL s with SAP BusinessObjects BI 4.0 Creator: Twitter: Blog: Pieter Verstraeten http://www.twitter.com/pverstraeten http://www.pieterverstraeten.com/blog Hi, Thanks for downloading

More information

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system 1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

More information

ÓÑÔ Ö Ø Ú ËØÙ Ý Ó ÌÛÓ ØÖÓÒÓÑ Ð ËÓ ØÛ Ö È Ò Ì Ø Ù Ñ ØØ Ò Ô ÖØ Ð ÙÐ ÐÐÑ ÒØ Ó Ø Ö ÕÙ Ö Ñ ÒØ ÓÖ Ø Ö Ó Å Ø Ö Ó Ë Ò Ò ÓÑÔÙØ Ö Ë Ò Ì ÍÒ Ú Ö ØÝ Ó Ù Ð Ò ½ ÌÓ ÅÙÑ Ò Ò ØÖ Ø Ì Ø ÓÑÔ Ö Ø Ú ØÙ Ý Ó ØÛÓ ÓÔ Ò ÓÙÖ ØÖÓÒÓÑ

More information

Ì È ÒÒ Ò ÌÖ Ò È Ö ËØÖÙØÙÖ ÒÒÓØ Ø ÓÒ Ó Ä Ö ÓÖÔÙ Æ ÒÛ Ò Ù Ù¹ ÓÒ ÓÙ Å ÖØ È ÐÑ Ö ÍÒ Ú Ö ØÝ Ó È ÒÒ ÝÐÚ Ò È Ð ÐÔ È ½ ½¼ ÍË ÜÙ Ò Û ÒÐ Òº ºÙÔ ÒÒº Ù Ü Ð Òº ºÙÔ ÒÒº Ù ÓÙ Ð Òº ºÙÔ ÒÒº Ù ÑÔ ÐÑ ÖÐ Òº ºÙÔ ÒÒº Ù ØÖ Ø

More information

Finding Near Rank Deficiency in Matrix Products

Finding Near Rank Deficiency in Matrix Products TR-CS-98-13 Finding Near Rank Deficiency in Matrix Products Michael Stewart December 1998 Joint Computer Science Technical Report Series Department of Computer Science Faculty of Engineering and Information

More information