MATH10212 Linear Algebra B Proof Problems
|
|
- Kimberly Arabella Burns
- 5 years ago
- Views:
Transcription
1 MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix C ab T is a matrix Prove that if > the Proof: Let C a 2 a a It is easy to see that rows of C are proportioal to b T with coefficiets a a that is a b T a 2 b T C a b T Therefore C by properties of ermiat 2 Let A M m are m matrices with m > Set C A T ; a it is a m m matrix Prove that C Proof: T is a m matrix where m > Therefore the system of homogeeous equatios T x has more variables tha equatios has a o-trivial solutio x ut the Cx ( A T ) x A ( t x ) A ( T ) A Hece C is a degeerate matrix C 3 I this problem we are workig with compoud 2 2 matrices of the form A C D where A C D M are matrices (a) Prove that O I ( ) ; I here I are the zero matrix the idetity matrix correspodigly of size (b) Express O A i terms of A Proof: (a) We kow that if we swap two differet rows i a ermiat the ermiat chages its sig Our matrix O I I is obtaied from the idetity matrix I 2 I I
2 MATH22 Liear Algebra Proof Problems 2 by movig each of first rows positios dowwards Obviously this requires 2 swaps of rows (b) Deote O I I I O ( ) 2 I ( ) 2 ( ) O I J I the O A O A A J A J A ( ) ( ) A by properties of ermiats of block-diagoal matrices 4 Let A be a 2 2 matrix which is similar oly to itself ot to ay other matrix Prove that A is a scalar matrix Proof: Sice A is similar oly to itself P AP A for all ivertible matrices P This is equivalet to AP P A Deote Take a b A P 2 The system of liear equatios yields Take a b 2 the the system yields [ 2 b c P a b a d Hece A is a scalar matrix [ a b a b Liear trasformatios Let T : V V be a liear trasformatio of a fiite dimesioal vector space V 5 Prove that if U < V is a vector subspace the T (U) { T (u) : u U } is a vector subspace of V 6 Prove that if T is oto it seds liearly idepedet sets of vectors to liearly idepedet sets 7 Prove that if T is ivertible U V the dim T (U) dim U 8 Let A be a matrix assume that A Prove that the map M M defied by X X A is a liear trasformatio 9 For give matrices A M 2 2 we defie the followig trasformatio R : M 2 2 M 2 2 R(X) AX Check that R is a liear trasformatio Prove that if R is oe-to-oe the both matrices A are ivertible Proof: Checkig the liearity of R is a direct expectio: R(X + Y ) A(X + Y ) (AX + AY ) AX + AY R(X) + R(Y ); R(cX) A(cX) (cax) cax cr(x) Assume that oe of the matrices A or is ot ivertible The it has ermiat
3 MATH22 Liear Algebra Proof Problems 3 the product AX has ermiat regardless of the choice of X Therefore all matrices i the image of trasformatio R have ermiat R caot be oto ut R is a liear trasformatio for liear trasformatios o fiite dimesioal vector spaces beig oto beig oe-to-oe are equivalet properties Hece R caot be oe-to-oe a cotradictio Aother solutio: Assume that R is oe-to-oe; as by part (ii) R is a liear trasformatio of a fiite-dimesioal vector space this is equivalet to assumig that R is oto Hece there is a matrix X such that AX I The A is ivertible with iverse X is ivertible with iverse AX Eigevalues eigevectors Prove that if a 2 2 matrix M has eigevalues the M ca be writte i the form a [c M d b for some umbers a b Hit: Use a matrix which cojugates M to a diagoal matrix Proof: M is cojugate to the diagoal matrix D say Deote the M A P DP p q P r s P P P P P [ s q r p s q p q r p r s qr qs pr ps [r s [ q p is i a desired form Aother solutio (shorter but more difficult): We start by showig that the claim is true for the most atural example of a matrix with eigevalues that is for D Ideed [ D Ay other such matrix M is similar to D so is equal to P DP hece M P DP P [ [ P It remais to deote by by P [ [ a b [ P [ Problem ca be stated i a much more geeral form solved without ivolvemet of eigevalues egevectors: Prove that if 2 2 matrix M is degeerate the M ca be writte i the form a [c M d b for some umbers a b Prove this statemet Symmetric orthogoal matrices 2 Recall that A are matrices the A A Prove that if A is a symmetric matrix is orthogoal the A is symmetric Proof: y defiitio of symmetric orthogo-
4 MATH22 Liear Algebra Proof Problems 4 al matrices Therefore A T A T ( A ) T ( A ) T ( T A ) T T A T ( T ) T T A A A Hece A is a symmetric matrix 3 A matrix A is atisymmetric if A T A Prove that if A is atisymmetric is a orthogoal matrix the A T is also atisymmetric Proof: y defiitio of atisymmetric orthogoal matrices Therefore A T A T ( A ) T ( A ) T ( T A ) T T A T ( T ) T T ( A) T A A ( A ) we ca suggest that the ermiat of a atisymmetric 3 3 matrix equals A proof is simple: we kow that that A A T for a atisymmetric matrix A this becomes A ( A) ( ) 3 A (we take out the scalar factor from each row of A that is three times!) therefore A A A (b) It clear that what matters i the proof above is that 3 is a odd umber So we wish to prove that It is easy: if is odd A is a atisymmetric matrix the A A A T for a atisymmetric matrix A this becomes A ( A) ( ) A (we take out the scalar factor from each of rows of A that is times!) Sice is odd ( ) hece A A A Hece A is a atisymmetric matrix 5 Prove that if a matrix is simultaeously triagular orthogoal the it is diagoal 4 Write several 3 3 atisymmetric matrices compute their ermiats (a) Make a cojecture about ermiats of atisymmetric 3 3 matrices prove it (b) Ca you geeralise your observatio to matrices of larger size prove it? Proof: (a) From a few simple examples such as 2 Proof: Assume that A is a upper triagular matrix of size that is all its etries below the diagoal equal (for lower triagular matrices the proof is the same with slight chage of words) Diagoal etries of A are its eigevalues; we kow that orthogoal matrices are ivertible; sice A is also orthogoal A is ivertible its eigevalues are ot equal We coclude that the diagoal etries a a 22 a are all o-zero Sice A is orthogoal its colums are orthogoal to each other The first
5 MATH22 Liear Algebra Proof Problems 5 colum of A is a a Colums a 2 a ca be orthogoal to a if their topmost etries a 2 a equal Repeatig the same argumet for a 2 we see that etries a 23 a 2 also equal Repeatig this process further we see that all elemets to the right of the diagoal equal ut the A is diagoal 6 Prove that the matrix A is ot similar to a symmetric matrix Ca A be similar to a atisymmetric matrix that is a matrix with the property A T A? Proof: Assume the cotrary let A is similar to a symmetric matrix S We kow that every symmetric matrix is similar (or cojugate) to a diagoal matrix; hece A is similar to some diagoal matrix D Sice A is triagular all eigevalues of A are diagoal etries equal Hece D is a diagoal matrix with all diagoal etries equal that is the idetity matrix ut the idetity matrix is similar oly to itself; that meas that A is the idetity matrix a obvious cotradictio A caot be similar to a atisymmetric matrix because A but atisymmetric matrices of odd size have zero ermiats
Inverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES 2012 Pearso Educatio, Ic. Theorem 8: Let A be a square matrix. The the followig statemets are equivalet. That is, for a give A, the statemets
More informationM 340L CS Homew ork Set 6 Solutions
1. Suppose P is ivertible ad M 34L CS Homew ork Set 6 Solutios A PBP 1. Solve for B i terms of P ad A. Sice A PBP 1, w e have 1 1 1 B P PBP P P AP ( ).. Suppose ( B C) D, w here B ad C are m matrices ad
More informationM 340L CS Homew ork Set 6 Solutions
. Suppose P is ivertible ad M 4L CS Homew ork Set 6 Solutios A PBP. Solve for B i terms of P ad A. Sice A PBP, w e have B P PBP P P AP ( ).. Suppose ( B C) D, w here B ad C are m matrices ad D is ivertible.
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More informationSymmetric Matrices and Quadratic Forms
7 Symmetric Matrices ad Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES SYMMERIC MARIX A symmetric matrix is a matrix A such that. A = A Such a matrix is ecessarily square. Its mai diagoal etries
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationLinearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation
Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationLecture 8: October 20, Applications of SVD: least squares approximation
Mathematical Toolkit Autum 2016 Lecturer: Madhur Tulsiai Lecture 8: October 20, 2016 1 Applicatios of SVD: least squares approximatio We discuss aother applicatio of sigular value decompositio (SVD) of
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationA Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
More informationIterative Techniques for Solving Ax b -(3.8). Assume that the system has a unique solution. Let x be the solution. Then x A 1 b.
Iterative Techiques for Solvig Ax b -(8) Cosider solvig liear systems of them form: Ax b where A a ij, x x i, b b i Assume that the system has a uique solutio Let x be the solutio The x A b Jacobi ad Gauss-Seidel
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationMatrix Algebra from a Statistician s Perspective BIOS 524/ Scalar multiple: ka
Matrix Algebra from a Statisticia s Perspective BIOS 524/546. Matrices... Basic Termiology a a A = ( aij ) deotes a m matrix of values. Whe =, this is a am a m colum vector. Whe m= this is a row vector..2.
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More informationChapter Unary Matrix Operations
Chapter 04.04 Uary atrix Operatios After readig this chapter, you should be able to:. kow what uary operatios meas, 2. fid the traspose of a square matrix ad it s relatioship to symmetric matrices,. fid
More informationCMSE 820: Math. Foundations of Data Sci.
Lecture 17 8.4 Weighted path graphs Take from [10, Lecture 3] As alluded to at the ed of the previous sectio, we ow aalyze weighted path graphs. To that ed, we prove the followig: Theorem 6 (Fiedler).
More information8. Applications To Linear Differential Equations
8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.
More informationDeterminants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)
5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More information24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS
24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS Corollary 2.30. Suppose that the semisimple decompositio of the G- module V is V = i S i. The i = χ V,χ i Proof. Sice χ V W = χ V + χ W, we have:
More informationTopics in Eigen-analysis
Topics i Eige-aalysis Li Zajiag 28 July 2014 Cotets 1 Termiology... 2 2 Some Basic Properties ad Results... 2 3 Eige-properties of Hermitia Matrices... 5 3.1 Basic Theorems... 5 3.2 Quadratic Forms & Noegative
More informationMATH 205 HOMEWORK #2 OFFICIAL SOLUTION. (f + g)(x) = f(x) + g(x) = f( x) g( x) = (f + g)( x)
MATH 205 HOMEWORK #2 OFFICIAL SOLUTION Problem 2: Do problems 7-9 o page 40 of Hoffma & Kuze. (7) We will prove this by cotradictio. Suppose that W 1 is ot cotaied i W 2 ad W 2 is ot cotaied i W 1. The
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationSingular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine
Lecture 11 Sigular value decompositio Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaie V1.2 07/12/2018 1 Sigular value decompositio (SVD) at a glace Motivatio: the image of the uit sphere S
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationSecond day August 2, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 1997, Plovdiv, BULGARIA Secod day August, 1997 Problems ad Solutios Let Problem 1. Let f be a C 3 (R) o-egative
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More information6. Kalman filter implementation for linear algebraic equations. Karhunen-Loeve decomposition
6. Kalma filter implemetatio for liear algebraic equatios. Karhue-Loeve decompositio 6.1. Solvable liear algebraic systems. Probabilistic iterpretatio. Let A be a quadratic matrix (ot obligatory osigular.
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig
More informationBasic Iterative Methods. Basic Iterative Methods
Abel s heorem: he roots of a polyomial with degree greater tha or equal to 5 ad arbitrary coefficiets caot be foud with a fiite umber of operatios usig additio, subtractio, multiplicatio, divisio, ad extractio
More informationGeometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT
OCTOBER 7, 2016 LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT Geometry of LS We ca thik of y ad the colums of X as members of the -dimesioal Euclidea space R Oe ca
More informationPROBLEM SET I (Suggested Solutions)
Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x
More informationThe Structure of Z p when p is Prime
LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationtoo many conditions to check!!
Vector Spaces Aioms of a Vector Space closre Defiitio : Let V be a o empty set of vectors with operatios : i. Vector additio :, v є V + v є V ii. Scalar mltiplicatio: li є V k є V where k is scalar. The,
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationMath E-21b Spring 2018 Homework #2
Math E- Sprig 08 Homework # Prolems due Thursday, Feruary 8: Sectio : y = + 7 8 Fid the iverse of the liear trasformatio [That is, solve for, i terms of y, y ] y = + 0 Cosider the circular face i the accompayig
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
More informationB = B is a 3 4 matrix; b 32 = 3 and b 2 4 = 3. Scalar Multiplication
MATH 37 Matrices Dr. Neal, WKU A m matrix A = (a i j ) is a array of m umbers arraged ito m rows ad colums, where a i j is the etry i the ith row, jth colum. The values m are called the dimesios (or size)
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationdenote the set of all polynomials of the form p=ax 2 +bx+c. For example, . Given any two polynomials p= ax 2 +bx+c and q= a'x 2 +b'x+c',
Chapter Geeral Vector Spaces Real Vector Spaces Example () Let u ad v be vectors i R ad k a scalar ( a real umber), the we ca defie additio: u+v, scalar multiplicatio: ku, kv () Let P deote the set of
More informationNotes The Incremental Motion Model:
The Icremetal Motio Model: The Jacobia Matrix I the forward kiematics model, we saw that it was possible to relate joit agles θ, to the cofiguratio of the robot ed effector T I this sectio, we will see
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationNAME: ALGEBRA 350 BLOCK 7. Simplifying Radicals Packet PART 1: ROOTS
NAME: ALGEBRA 50 BLOCK 7 DATE: Simplifyig Radicals Packet PART 1: ROOTS READ: A square root of a umber b is a solutio of the equatio x = b. Every positive umber b has two square roots, deoted b ad b or
More informationA brief introduction to linear algebra
CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad
More informationMath 4707 Spring 2018 (Darij Grinberg): homework set 4 page 1
Math 4707 Sprig 2018 Darij Griberg): homewor set 4 page 1 Math 4707 Sprig 2018 Darij Griberg): homewor set 4 due date: Wedesday 11 April 2018 at the begiig of class, or before that by email or moodle Please
More information(VII.A) Review of Orthogonality
VII.A Review of Orthogoality At the begiig of our study of liear trasformatios i we briefly discussed projectios, rotatios ad projectios. I III.A, projectios were treated i the abstract ad without regard
More informationA Note on Matrix Rigidity
A Note o Matrix Rigidity Joel Friedma Departmet of Computer Sciece Priceto Uiversity Priceto, NJ 08544 Jue 25, 1990 Revised October 25, 1991 Abstract I this paper we give a explicit costructio of matrices
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationLecture 23 Rearrangement Inequality
Lecture 23 Rearragemet Iequality Holde Lee 6/4/ The Iequalities We start with a example Suppose there are four boxes cotaiig $0, $20, $50 ad $00 bills, respectively You may take 2 bills from oe box, 3
More informationMath 778S Spectral Graph Theory Handout #3: Eigenvalues of Adjacency Matrix
Math 778S Spectral Graph Theory Hadout #3: Eigevalues of Adjacecy Matrix The Cartesia product (deoted by G H) of two simple graphs G ad H has the vertex-set V (G) V (H). For ay u, v V (G) ad x, y V (H),
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More informationMath 203A, Solution Set 8.
Math 20A, Solutio Set 8 Problem 1 Give four geeral lies i P, show that there are exactly 2 lies which itersect all four of them Aswer: Recall that the space of lies i P is parametrized by the Grassmaia
More informationRegn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,
. Sectio-A cotais 30 Multiple Choice Questios (MCQ). Each questio has 4 choices (a), (b), (c) ad (d), for its aswer, out of which ONLY ONE is correct. From Q. to Q.0 carries Marks ad Q. to Q.30 carries
More informationMathematics Review for MS Finance Students Lecture Notes
Mathematics Review for MS Fiace Studets Lecture Notes Athoy M. Mario Departmet of Fiace ad Busiess Ecoomics Marshall School of Busiess Uiversity of Souther Califoria Los Ageles, CA 1 Lecture 1.1: Basics
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationChapter 6: Determinants and the Inverse Matrix 1
Chapter 6: Determiats ad the Iverse Matrix SECTION E pplicatios of Determiat By the ed of this sectio you will e ale to apply Cramer s rule to solve liear equatios ermie the umer of solutios of a give
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationThe Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005
The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationHomework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is
Homewor 3 Chapter 5 pp53: 3 40 45 Chapter 6 p85: 4 6 4 30 Use combiatorial reasoig to prove the idetity 3 3 Proof Let S be a set of elemets ad let a b c be distict elemets of S The umber of -subsets of
More informationAfter the completion of this section the student should recall
Chapter III Liear Algebra September 6, 7 6 CHAPTER III LINEAR ALGEBRA Objectives: After the completio of this sectio the studet should recall - the cocept of vector spaces - the operatios with vectors
More informationState Space Representation
Optimal Cotrol, Guidace ad Estimatio Lecture 2 Overview of SS Approach ad Matrix heory Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore State Space Represetatio Prof.
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationCardinality Homework Solutions
Cardiality Homework Solutios April 16, 014 Problem 1. I the followig problems, fid a bijectio from A to B (you eed ot prove that the fuctio you list is a bijectio): (a) A = ( 3, 3), B = (7, 1). (b) A =
More informationCov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n.
CS 189 Itroductio to Machie Learig Sprig 218 Note 11 1 Caoical Correlatio Aalysis The Pearso Correlatio Coefficiet ρ(x, Y ) is a way to measure how liearly related (i other words, how well a liear model
More informationCourse : Algebraic Combinatorics
Course 18.312: Algebraic Combiatorics Lecture Notes # 18-19 Addedum by Gregg Musier March 18th - 20th, 2009 The followig material ca be foud i a umber of sources, icludig Sectios 7.3 7.5, 7.7, 7.10 7.11,
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationMath 128A: Homework 1 Solutions
Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that
More information