Honours Algebra 2, Assignment 8
|
|
- Kevin May
- 5 years ago
- Views:
Transcription
1 Honours Algebra, Assignment 8 Jamie Klassen and Michael Snarski April 10, 01 Question 1. Let V be the vector space over the reals consisting of polynomials of degree at most n 1, and let x 1,...,x n be n distinct real numbers. a) Show that the functionals l 1,...,l n on V defined by l j (p(x)) = p(x j ) 1 j n form a basis for the dual space V. b) (Polynomial interpolation). Given real numbers y 1,...,y n, show that there is a unique polynomial of degree at most n 1 satisfying p(x j ) = y j for all 1 j n, and that it is given by the formula p(x) = p(x 1 )p 1 (x) + + p(x n )p n (x) where (p 1 (x),..., p n (x)) is the basis of V which is dual to (l 1,...,l n ), i.e., satisfies (p 1,...,p n) = (l 1,...,l n ). c) Write down a formula for the basis (p 1,...,p n ). Solution (a) Since we are dealing with a finite-dimensional space, dimv = dimv = n, and so it suffices to show the n vectors l 1,...,l n are linearly independent. Suppose that l = n i=1 α il i = 0; we need to show α i = 0 for all i, so pick any α j. Since l is the zero operator, l(p(x)) = 0 for all p(x) V. Consider the polynomial p j (x) = i j (x x i). We have ( n ) l(p j (x)) = α i l i x i ) i j(x i=1 = α 1 (x 1 x 1 )...(x 1 x n ) + + α j (x j x 1 )...(x j x n ) + + α n (x n x 1 )...(x n x n ) = α j p j (x j ) = 0. Since all the numbers x 1,...,x n are distinct, p j (x j ) 0 by construction, whence it follows that α j = 0. Since α j was arbitrary, we conclude that α i = 0 for i = 1,...,n, and we are done. (b) The polynomial is given, so there is no need to prove existence as many of you did. To prove uniqueness, suppose there are two polynomials p(x) and q(x) of degree at most n 1 satisfying p(x j ) = q(x j ) = y j for j = 1,...,n. The polynomial p(x) q(x) has degree at most n 1 as well and has n roots, so because we are over a field, p(x) q(x) 0. To see that the given polynomial n p(x) = p i (x)p(x i ) = p 1 (x 1 ) + + p n (x n ) i=1 1
2 works, simply plug in x j and find p(x j ) = n p i (x j )p(x i ) = p j (x j )p(x j ) = p(x j ). i=1 (c) We need only write down the formula: p j (x) = i j x x i x j x i. (This is a little confusing, so write it out and you ll see why it works.) Question. Proposition 7..1 of Eyal Goren s notes shows that there is a natural injective linear transformation from V to V, the dual of the dual of V. It then shows furthermore that this map is an isomorphism when V is finite-dimensional. Read the proof of Proposition 7..1 carefully and note the crucial role played by the finite-dimensionality assumption. (You do not need to write anything, yet...) ow, give an example to show that the natural map V V need not be surjective in general, if V is not assumed to be finite dimensional. Solution Some of you discussed some facts of functional analysis, sequence spaces and continuous duals, which are cool; however, it is important to know that the duals you discuss in functional analysis are proper subspaces of the algebraic dual, consisting only of those functionals which are continuous or have finite operator norm. On the other hand, several of you exhibited the example of V = Z [x], the vector space of polynomials with coefficients in Z, proving that the natural map is not surjective by giving a countable basis for V and showing that V must have an uncountable basis. I will generalize the above approaches and show you the cool-but-not-very-often-proven fact that in infinite dimensions, the natural injection is never surjective. The proof: The axiom of choice will rear its ugly head a few times most notably as Hamel s basis theorem, that every vector space has a basis or more generally that any linearly independent set can be extended to a basis. So let {v i } i I be a basis for V (of course I is infinite by assumption), and let B be its dual basis (v i (v j) = δ ij extended by linearity). Extend B to a basis D for V. ow define a linear functional ψ : V F by ψ(f) = 1 for all f D extended by linearity. Let φ : V V be the natural injection φ(v)(f) = f(v). We will show that ψ / image(φ) = { k K a k φ(v k ) : a k F, K I is a finite subset}. To see this, suppose to the contrary that ψ = k K a kφ(v k ) image(φ) for some finite K I. Since I is infinite I \ K, so let j I \ K be arbitrary. Then vj D is an element of the dual basis so ψ(vj ) = 1 and v j (v k) = 0 for all k K, so 1 = ψ(vj) = a k φ(v k )(vj) = a k vj(v k ) = 0, k K k K a contradiction. Thus ψ V \ image(φ) so φ is not surjective. Question 3. Let H = R i, j, k be the usual ring of quaternions. (Recall that a typical quaternion is an element of the form z = a + bi + cj + dk, a, b, c, d R.)
3 Recall the trace and norm functions on H defined by trace(z) = a = z + z, norm(z) = a + b + c + d = z z, where z = a bi cj dk. Show that H, equipped with the rule x, y := trace(xȳ) is a real inner product space. Solution The fact that H is a real vector space is essentially trivial; if {e 1,...,e 4 } is the standard basis for R 4, then we can biject H to R 4 by 1 e 1, i e, j e 3, k e 4 and the vector space axioms follow immediately (strictly speaking one should state that we are taking the formal addition and scalar multiplication given by the ring structure as our operations). To see that the given rule is an inner product, we establish the identity xy = xȳ; letting x = a+bi+cj+dk, y = a +b i+c j+d k we have, using the identity i = j = k = ijk = 1: xy = (a + bi + cj + dk)(a + b i + c j + d k) = aa bb cc dd + (ab ba + cd dc )i + (ac bd + ca + db )j + (ad + bc cb + da )k = aa bb cc dd (ab ba + cd dc )i (ac bd + ca + db )j (ad + bc cb + da )k = (a bi cj dk)(a b i c j d k) = xȳ. Furthermore, trace(xy) = aa bb cc dd = a a b b c c d d = trace(yx) and and for α R trace(x + y) = (a + a ) = a + a = trace(x) + trace(y) trace(αx) = (αa) = α(a) = α trace(x). ow we check the axioms by hand: i. Well-defined: Just for thoroughness, the trace of a quaternion is always a real number. ii. Symmetry: We have in general that x = a bi cj dk = a + bi + cj + dk = x so x, y = trace(xȳ) = xȳ + xȳ = xy + xy = trace( xy) = trace(y x) = y, x. iii. Positive definiteness: We have x, x = trace(x x) = (a + b + c + d ) 0 since squares are nonnegative, with equality if and only if a = b = c = d = 0; i.e. if and only if x = 0. iv. Bilinearity: It suffices to check linearity in the first argument since we have already established 3
4 symmetry. We have for α, β R and x, y, z H, as required. αx + βy, z = trace((αx + βy) z) = trace(αx z + βy z) = trace(αx z) + trace(βy z) = α trace(x z) + β trace(y z) = α x, z + β y, z Question 4. ote that H can also be viewed as a complex vector space by viewing the elements of the form a+bi as complex numbers and using the multiplication in H to define the scalar multiplication. Is H equipped with the pairing above a Hermitian space? Explain. Solution ope. In a Hermitian inner product space, we require αx, y = αxy (physicists use linearity in the second argument, x, αy = α x, y ; I ve come to trust physics on all things notation). Let x, y H and α C. We have: α x, y = α(xy + xy) = αxy + αxy αx, y = αxy + αxy so unless α = α (i.e. α R, we will not have equality for all x, y H, and so this is not a Hermitian space. Question 5. Let V be the vector space of real-valued functions on the interval [, π] equipped with the inner product f, g := f(t)g(t) dt. (marker s note: we must assume that this inner product is always well-defined, perhaps by restricting our attention to those functions which are Riemann-integrable on [, π]) Let W be the subspace of functions spanned by f 0 = 1/ π, f j (t) := 1/ π cos(jt), with 1 j and g j (t) := 1/ π sin(jt) with 1 j. Show that f 0, f 1,...,f, g 1,...,g is an orthonormal basis for W. Given f V, give a formula for the coefficients of the function a 0 f 0 + a 1 f a f + b 1 g b g in W which best approximates f. Compute these coefficients in the case of the function f(x) = x. Solution f 0, f 1,...,f, g 1,...,g is a basis for V by construction; more precisely, it spans by definition, and the functions involved will be seen to be linearly independent by showing they are mutually orthogonal. The proof that this set is orthonormal is a bit long and tedious, involving a bunch of integration tricks it was marked somewhat generously. If you had trouble with it (most of you did not), use the identities sin(a)cos(b) = 1 (sin(a + b) + sin(a b)), sin(a)sin(b) = 1 (cos(a + b) cos(a b)), and cos(a)cos(b) = 1 (cos(a + b) + cos(a b)) 4
5 to ease the process. Of course, given an arbitrary f V, the orthogonal projection onto W will yield the vector in W closest (with respect to the norm metric induced by our inner product) to V. This means a 0 = 1 π f(t)dt, a i = 1 π f(t)cos(it)dt, b i = 1 π f(t)sin(it)dt, 1 i. Computing in the particular case of f(x) = x gives a 0 = a i = 0 for all i since x and xcos(ix) are odd functions, and b j = 1 π t sin(jt)dt = sin(jt) jt cos(jt) π j = 1 sin(jπ) jπ cos(jπ) sin( jπ) jπ cos( jπ) t= π j = π( 1) j+1. j Question 6. Let V be the space of polynomials of degree equipped with the inner product f, g = 1 0 f(t)g(t)dt. Find an orthonormal basis for V, by applying the Gramm-Schmidt procedure to the ordered basis (1, x, x ). Solution An exercise in computation, something the entire honours math stream needs more of. Let v 1 = 1, and note that v 1, v 1 = 1 0 dt = 1, so our first vector is already normalized. v = x x,1 1, 1 1 = x = x 1. We will have to normalize v, but we do that last as scaling it will simply make computation more difficult, and v, v = v will come out in the next computation anyway. 1 0 tdt v 3 = x x, 1 1, 1 1 x, x 1/ (x 1/) x 1/, x 1/ ( = x t3 t dt ) 1 0 t t dt(x 1 ) = x 1 ( 1/1 3 1/1 (x 1 ) ) = x x ormalizing, v v = ( 1 x 1 ) = 3x 3 v 3 v 3 = ( 180 x x + 1 ) = 6 5x so our orthonormal basis is {1, 3x 3, 6 5x }. Question 7. Let (x 1, x,...,x ) and (y 1,...,y ) be sequences of real numbers. Fix an integer k less than. In general, there need not be a polynomial p of degree k such that p(x j ) = y j for j = 1,...,. The next-best thing one might ask for is a polynomial p of degree k for which the quantity (p(x 1 ) y 1 ) + (p(x ) y ) + + (p(x ) y ) 5
6 is minimised. Describe an approach that would produce such a p. In the case k = 1, give a formula for the coefficients a, b of p(x) = ax + b in terms of x 1,...,x, y 1,...,y. Solution B: Many of you solved this problem with calculus methods, obtaining the correct solution I gave full marks for this when it satisfactorily solved the problem. It s debatable whether this method is easier, but it is certainly less motivated, given the inner product space type solution, which I present here. Let V be the vector space of polynomials of degree 1, and let W be the subspace of polynomials of degree k. From question 1 there is a polynomial q V satisfying q(x j ) = y j for j = 1,...,. Define an inner product on V by f, g := f(x 1 )g(x 1 ) + f(x )g(x ) + + f(x )g(x ). Then the quantity in the question is p q = p q, p q, and the p minimizing this quantity can be found by using the Gramm-Schmidt process to produce an orthonormal basis for W and computing the orthogonal projection of q onto this basis. For k = 1 we start with the basis {1, x} for W, and Gramm-Schmidt gives 1, as an orthonormal basis. Projecting gives x 1 x k x k 1 ( n x k) p = 1, q 1 x 1 + x k x 1 ( x k 1 ), q x k x ( k x k 1 k) x ( = 1 x ky k 1 ( x k)( ) ( y k) x 1 ) x k y k + x k 1 k) x = x ky k 1 ( x k)( y k) x k 1 x + k) 1 y k x ky k 1 ( x k)( y k) 1 x x k 1 ) x k x k so a = x ky k 1 ( x k)( y k) x k 1 k) x and b = 1 y k x ky k 1 ( x k)( y k) 1 x k 1 ) x k x k. Another nice solution to this problem comes from considering a polynomial as the vector of its coefficients, letting A be the transformation which evaluates a polynomial at x 1,...,x and minimizing the Euclidean distance between (y 1,...,y ) and A(p), which is really just a reformulation of the solution above. 6
Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationMath 110, Spring 2015: Midterm Solutions
Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More informationMODEL ANSWERS TO THE FIRST HOMEWORK
MODEL ANSWERS TO THE FIRST HOMEWORK 1. Chapter 4, 1: 2. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above
More informationMATH 423 Linear Algebra II Lecture 12: Review for Test 1.
MATH 423 Linear Algebra II Lecture 12: Review for Test 1. Topics for Test 1 Vector spaces (F/I/S 1.1 1.7, 2.2, 2.4) Vector spaces: axioms and basic properties. Basic examples of vector spaces (coordinate
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationMIDTERM I LINEAR ALGEBRA. Friday February 16, Name PRACTICE EXAM SOLUTIONS
MIDTERM I LIEAR ALGEBRA MATH 215 Friday February 16, 2018. ame PRACTICE EXAM SOLUTIOS Please answer the all of the questions, and show your work. You must explain your answers to get credit. You will be
More informationThere are two things that are particularly nice about the first basis
Orthogonality and the Gram-Schmidt Process In Chapter 4, we spent a great deal of time studying the problem of finding a basis for a vector space We know that a basis for a vector space can potentially
More informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More informationMODEL ANSWERS TO HWK #7. 1. Suppose that F is a field and that a and b are in F. Suppose that. Thus a = 0. It follows that F is an integral domain.
MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0
More informationInner product spaces. Layers of structure:
Inner product spaces Layers of structure: vector space normed linear space inner product space The abstract definition of an inner product, which we will see very shortly, is simple (and by itself is pretty
More informationLinear Models Review
Linear Models Review Vectors in IR n will be written as ordered n-tuples which are understood to be column vectors, or n 1 matrices. A vector variable will be indicted with bold face, and the prime sign
More informationREAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS
REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS The problems listed below are intended as review problems to do before the final They are organied in groups according to sections in my notes, but it is not
More information2: LINEAR TRANSFORMATIONS AND MATRICES
2: LINEAR TRANSFORMATIONS AND MATRICES STEVEN HEILMAN Contents 1. Review 1 2. Linear Transformations 1 3. Null spaces, range, coordinate bases 2 4. Linear Transformations and Bases 4 5. Matrix Representation,
More informationSOLUTIONS TO EXERCISES FOR MATHEMATICS 133 Part 1. I. Topics from linear algebra
SOLUTIONS TO EXERCISES FOR MATHEMATICS 133 Part 1 Winter 2009 I. Topics from linear algebra I.0 : Background 1. Suppose that {x, y} is linearly dependent. Then there are scalars a, b which are not both
More informationLINEAR ALGEBRA MICHAEL PENKAVA
LINEAR ALGEBRA MICHAEL PENKAVA 1. Linear Maps Definition 1.1. If V and W are vector spaces over the same field K, then a map λ : V W is called a linear map if it satisfies the two conditions below: (1)
More informationMath 61CM - Solutions to homework 6
Math 61CM - Solutions to homework 6 Cédric De Groote November 5 th, 2018 Problem 1: (i) Give an example of a metric space X such that not all Cauchy sequences in X are convergent. (ii) Let X be a metric
More informationLinear Algebra Lecture Notes-II
Linear Algebra Lecture Notes-II Vikas Bist Department of Mathematics Panjab University, Chandigarh-64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered
More informationLinear Algebra Notes. Lecture Notes, University of Toronto, Fall 2016
Linear Algebra Notes Lecture Notes, University of Toronto, Fall 2016 (Ctd ) 11 Isomorphisms 1 Linear maps Definition 11 An invertible linear map T : V W is called a linear isomorphism from V to W Etymology:
More informationMath 1060 Linear Algebra Homework Exercises 1 1. Find the complete solutions (if any!) to each of the following systems of simultaneous equations:
Homework Exercises 1 1 Find the complete solutions (if any!) to each of the following systems of simultaneous equations: (i) x 4y + 3z = 2 3x 11y + 13z = 3 2x 9y + 2z = 7 x 2y + 6z = 2 (ii) x 4y + 3z =
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More information2.2. Show that U 0 is a vector space. For each α 0 in F, show by example that U α does not satisfy closure.
Hints for Exercises 1.3. This diagram says that f α = β g. I will prove f injective g injective. You should show g injective f injective. Assume f is injective. Now suppose g(x) = g(y) for some x, y A.
More informationDavid Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent
Chapter 5 ddddd dddddd dddddddd ddddddd dddddddd ddddddd Hilbert Space The Euclidean norm is special among all norms defined in R n for being induced by the Euclidean inner product (the dot product). A
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationMATH Linear Algebra
MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization
More informationLinear Algebra Lecture Notes-I
Linear Algebra Lecture Notes-I Vikas Bist Department of Mathematics Panjab University, Chandigarh-6004 email: bistvikas@gmail.com Last revised on February 9, 208 This text is based on the lectures delivered
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More informationMath Real Analysis II
Math 4 - Real Analysis II Solutions to Homework due May Recall that a function f is called even if f( x) = f(x) and called odd if f( x) = f(x) for all x. We saw that these classes of functions had a particularly
More informationTypical Problem: Compute.
Math 2040 Chapter 6 Orhtogonality and Least Squares 6.1 and some of 6.7: Inner Product, Length and Orthogonality. Definition: If x, y R n, then x y = x 1 y 1 +... + x n y n is the dot product of x and
More informationLinear Algebra, Summer 2011, pt. 3
Linear Algebra, Summer 011, pt. 3 September 0, 011 Contents 1 Orthogonality. 1 1.1 The length of a vector....................... 1. Orthogonal vectors......................... 3 1.3 Orthogonal Subspaces.......................
More informationChapter 5. Basics of Euclidean Geometry
Chapter 5 Basics of Euclidean Geometry 5.1 Inner Products, Euclidean Spaces In Affine geometry, it is possible to deal with ratios of vectors and barycenters of points, but there is no way to express the
More informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More information(v, w) = arccos( < v, w >
MA322 F all206 Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: Commutativity:
More informationSolutions to odd-numbered exercises Peter J. Cameron, Introduction to Algebra, Chapter 2
Solutions to odd-numbered exercises Peter J Cameron, Introduction to Algebra, Chapter 1 The answers are a No; b No; c Yes; d Yes; e No; f Yes; g Yes; h No; i Yes; j No a No: The inverse law for addition
More informationLINEAR ALGEBRA (PMTH213) Tutorial Questions
Tutorial Questions The tutorial exercises range from revision and routine practice, through lling in details in the notes, to applications of the theory. While the tutorial problems are not compulsory,
More informationInner Product Spaces
Inner Product Spaces Introduction Recall in the lecture on vector spaces that geometric vectors (i.e. vectors in two and three-dimensional Cartesian space have the properties of addition, subtraction,
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationMATH1050 Greatest/least element, upper/lower bound
MATH1050 Greatest/ element, upper/lower bound 1 Definition Let S be a subset of R x λ (a) Let λ S λ is said to be a element of S if, for any x S, x λ (b) S is said to have a element if there exists some
More information(v, w) = arccos( < v, w >
MA322 F all203 Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v,
More informationMTH 503: Functional Analysis
MTH 53: Functional Analysis Semester 1, 215-216 Dr. Prahlad Vaidyanathan Contents I. Normed Linear Spaces 4 1. Review of Linear Algebra........................... 4 2. Definition and Examples...........................
More informationMATH Linear Algebra
MATH 4 - Linear Algebra One of the key driving forces for the development of linear algebra was the profound idea (going back to 7th century and the work of Pierre Fermat and Rene Descartes) that geometry
More informationMath 396. Quotient spaces
Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition
More informationMath Topology II: Smooth Manifolds. Spring Homework 2 Solution Submit solutions to the following problems:
Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 2 Solution Submit solutions to the following problems: 1. Let H = {a + bi + cj + dk (a, b, c, d) R 4 }, where i 2 = j 2 = k 2 = 1, ij = k,
More informationV (v i + W i ) (v i + W i ) is path-connected and hence is connected.
Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationHilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality
(October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are
More information1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces.
1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces. The set of n-tuples of real numbers is denoted by R n. Suppose that a is a real number
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 994, 28. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More information(v, w) = arccos( < v, w >
MA322 Sathaye Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v
More informationSETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
More informationFurther Mathematical Methods (Linear Algebra)
Further Mathematical Methods (Linear Algebra) Solutions For The Examination Question (a) To be an inner product on the real vector space V a function x y which maps vectors x y V to R must be such that:
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationRings. EE 387, Notes 7, Handout #10
Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationUMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM
UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM YUFEI ZHAO In this talk, I want give some examples to show you some linear algebra tricks for the Putnam. Many of you probably did math contests in
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW When we define a term, we put it in boldface. This is a very compressed review; please read it very carefully and be sure to ask questions on parts you aren t sure of. x 1 WedenotethesetofrealnumbersbyR.
More informationInner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:
Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v
More informationExercises for Unit I I (Vector algebra and Euclidean geometry)
Exercises for Unit I I (Vector algebra and Euclidean geometry) I I.1 : Approaches to Euclidean geometry Ryan : pp. 5 15 1. What is the minimum number of planes containing three concurrent noncoplanar lines
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN-3 = 978--68-78376-2,
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationA Primer in Econometric Theory
A Primer in Econometric Theory Lecture 1: Vector Spaces John Stachurski Lectures by Akshay Shanker May 5, 2017 1/104 Overview Linear algebra is an important foundation for mathematics and, in particular,
More informationHonors Algebra II MATH251 Course Notes by Dr. Eyal Goren McGill University Winter 2007
Honors Algebra II MATH251 Course Notes by Dr Eyal Goren McGill University Winter 2007 Last updated: April 4, 2014 c All rights reserved to the author, Eyal Goren, Department of Mathematics and Statistics,
More informationFurther Mathematical Methods (Linear Algebra)
Further Mathematical Methods (Linear Algebra) Solutions For The 2 Examination Question (a) For a non-empty subset W of V to be a subspace of V we require that for all vectors x y W and all scalars α R:
More informationHOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.
HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p
More information4.3 - Linear Combinations and Independence of Vectors
- Linear Combinations and Independence of Vectors De nitions, Theorems, and Examples De nition 1 A vector v in a vector space V is called a linear combination of the vectors u 1, u,,u k in V if v can be
More informationProf. Ila Varma HW 8 Solutions MATH 109. A B, h(i) := g(i n) if i > n. h : Z + f((i + 1)/2) if i is odd, g(i/2) if i is even.
1. Show that if A and B are countable, then A B is also countable. Hence, prove by contradiction, that if X is uncountable and a subset A is countable, then X A is uncountable. Solution: Suppose A and
More informationCHAPTER II HILBERT SPACES
CHAPTER II HILBERT SPACES 2.1 Geometry of Hilbert Spaces Definition 2.1.1. Let X be a complex linear space. An inner product on X is a function, : X X C which satisfies the following axioms : 1. y, x =
More informationSPECTRAL THEORY EVAN JENKINS
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for
More informationGeneral Inner Product & Fourier Series
General Inner Products 1 General Inner Product & Fourier Series Advanced Topics in Linear Algebra, Spring 2014 Cameron Braithwaite 1 General Inner Product The inner product is an algebraic operation that
More informationSupplementary Material for MTH 299 Online Edition
Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think
More informationName (print): Question 4. exercise 1.24 (compute the union, then the intersection of two sets)
MTH299 - Homework 1 Question 1. exercise 1.10 (compute the cardinality of a handful of finite sets) Solution. Write your answer here. Question 2. exercise 1.20 (compute the union of two sets) Question
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationAlgebraic structures I
MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationSection 7.5 Inner Product Spaces
Section 7.5 Inner Product Spaces With the dot product defined in Chapter 6, we were able to study the following properties of vectors in R n. ) Length or norm of a vector u. ( u = p u u ) 2) Distance of
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationTranspose & Dot Product
Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationMath 300: Final Exam Practice Solutions
Math 300: Final Exam Practice Solutions 1 Let A be the set of all real numbers which are zeros of polynomials with integer coefficients: A := {α R there exists p(x) = a n x n + + a 1 x + a 0 with all a
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationINNER PRODUCT SPACE. Definition 1
INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of
More information