# Interpolating Polynomials Handout March 7, 2012

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) in the plane For this, we must assume that the x-coordinates x 1,x 2,,x n are distinct (Otherwise if x i = x j, it is clearly impossible to fit a function exactly to the data unless also y i = y j ) It is always possible to fit a polynomial function to the data exactly if one allows polynomials of sufficiently high degree The good news is that there is a polynomial f(x) of degree < n which fits the data exactly It is not hard to see that such a polynomial must be unique For suppose y = f(x) and y = g(x) are two polynomials of degree < n, both of which fit the n data points Then the difference polynomial f(x) g(x) has degree < n and vanishes at x 1,x 2,,x n But this implies that f(x) g(x) =, ie f(x) = g(x) Theorem 1 Let (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) be ordered pairs of elements of F, and suppose that the values x 1,x 2,,x n are distinct Then there exists a unique polynomial p(x) F[x] of degree < n such that p(x i ) = y i for i = 1,2,,n We have seen why f(x) must be unique, but not yet why it exists We will in fact give two different proofs of this fact Let us write p(x) = a + a 1 x + a 2 x a n 1 x n 1 We determine the n unknown coefficients a,,a n 1 F using the n linear equations a + a 1 x 1 + a 2 x a n 1 x n 1 1 = y 1, a + a 1 x 2 + a 2 x a n 1 x n 1 2 = y 2, a + a 1 x 3 + a 2 x a n 1 x n 1 3 = y 3, a + a 1 x n + a 2 x 2 n + + a n 1 x n 1 n = y n In matrix form, this linear system is expressed as AX = B, where 1

2 1 x 1 x 2 1 x n x 2 x 2 2 x n 1 2 A = 1 x 3 x 2 3 x n 1 3, X = 1 x n x 2 n x n 1 n a a 1 a 2 a n 1, B = The matrix A shown is a very special n n matrix, known as a Vandermonde matrix The linear system AX = B has a unique solution, namely X = A 1 B, if and only if det(a) So to prove Theorem 2, what we really need to show is that det(a) It is not hard to compute the case n = 1: [ 1 x1 A = 1 x 2 ], det(a) = x 2 x 1 and the case n = 2: 1 x 1 x 2 1 A = 1 x 2 x 2 2, det(a) = (x 2 x 1 )(x 3 x 1 )(x 3 x 2 ) 1 x 3 x 2 3 The general expression for det(a) is given by the following: y 1 y 2 y 3 y n Lemma 2 The determinant of the n n Vandermonde matrix A as above, is given by det(a) = (x j x i ) 1 i<j n Here the notation means product (similar to the notation meaning sum, with which you should be familiar) Note that the cases n = 1, 2 of Lemma 2 give exactly the determinants we computed above Rather than give a complete proof of Lemma 2, we give the ideas of the proof Note that det(a) is a polynomial h(x 1,x 2,,x n ) Also, expanding det(a) using the definition of determinant, each of the n! terms is a monomial of degree (n 1) = n(n 1)/2, so deg (h) = n(n 1)/2 Also, this polynomial vanishes whenever x i = x j where i < j This implies that the product of all of the linear factors (x j x i ) divides h(x 1,x 2,,x n ), so h(x 1,x 2,,x n ) = c(x 1,x 2,,x n ) (x j x i ) 2 1 i<j n

3 for some polynomial c(x 1,x 2,,x n ) F[x 1,x 2,,x n ] However, the number of such linear factors x j x i with 1 i < j n equals the binomial coefficient ( ) n n! n(n 1) = = 2 2!(n 2)! 2 By comparing degrees, we must have degc(x 1,,x n ) =, ie we have a constant polynomial c(x 1,,x n ) = c F To determine the constant c, observe that the main diagonal of A gives a term x 2 x 2 3x 3 4 x n 1 n in h(x 1,,x n ) A similar term in the expansion of the product shows that c = 1 This proves Lemma 2 Now it is easy to prove Theorem 1 from Lemma 2 Since by the hypothesis of Theorem 1, we assume the x i s to be distinct, Lemma 2 gives det(a) Thus A is invertible and the system AX = B has a unique solution X = A 1 B The entries of the column vector X give the coefficients of the polynomial p(x) Now let us give an alternative proof of Theorem 1 Let V be the vector space of all polynomials in x of degree < n with coefficients in F, ie V = {a + a 1 x + a 2 x a n 1 x n 1 : a,a 1,,a n 1 F } Every vector p(x) V can be uniquely expressed as a linear combination of the vectors 1,x,x 2,,x n 1, viz p(x) = a 1 + a 1 x + a 2 x a n 1 x n 1 where a,a 1,,a n 1 F This means that {1,x,x 2,,x n 1 } is a basis for V It is not the only basis (soon we will exhibit others) but all bases have the same size This size (ie n) is by definition the dimension of V ie Consider another vector space W consisting of all column vectors of length n over F, W = b 1 b 2 b n : b 1,b 2,,b n F Every vector w W can be uniquely expressed as a linear combination of the n vectors 1, 1,, ; 3 1

4 namely, 1 w = b 1 + b b n 1 So we have a set of n column vectors forming a basis for W This means that W also has dimension n Now suppose x 1,x 2,,x n F are distinct, and consider the map T : V W, T(p(x)) = p(x 1 ) p(x 2 ) p(x n ) It is easy to see that T is a linear transformation, ie T(ap(x)+bq(x)) = at(p(x))+bt(q(x)) for all a, b F and all p(x), q(x) V The Fundamental Theorem of Linear Algebra says that dim ( null space of T ) + dim ( image of T ) = dimv = n Now the null space of T consists of all polynomials p(x) V such that T(p(x)) = p(x 1 ) p(x 2 ) p(x n ) = The only polynomial satisfying this condition is p(x) =, since a nonzero polynomial of degree less than n cannot have n distinct roots So the null space of T is zero (the subspace consisting of just the zero vector) This subspace has dimension Now the image of T is a subspace of W of dimension n But W itself has dimension n So the image of T must consist of all of W, ie the linear transformation T : V W is onto It is also one-to-one since its null space is zero (These are among the most basic facts from Linear Algebra; if they are not familiar, you should review them) Now the (second) proof of Theorem 1 is clear: Given w = y 1 y 2 y n 4 W,

5 there exists a unique p(x) V such that T(p(x)) = w The existence of p(x) V comes from the fact that T is onto; and the uniqueness we have seen before (and is equivalent to the fact that T is one-to-one, since its null space is zero) As an example, consider the problem of finding the unique polynomial p(x) = a+bx+ cx 2 of degree < 3 through the three points (,7), (1,6), (2,9) This gives a linear system a = 7, a + b + c = 6, a + 2b + 4c = 9 Solving this system by elementary row operations on the corresponding augmented matrix (which is faster than inverting the 3 3 coefficient matrix) gives the unique solution a=7, b= 3, c =2, and so the unique polynomial of degree < 3 passing through the three points given is p(x) = 7 3x + 2x 2 (You should check that this polynomial does indeed pass through the three points given) Working with n n matrices, while possible in principle, becomes impractical for large values of n We now give two other approaches to determining the interpolating polynomial p(x), using more direct formulas which minimize the requirements on computational resources Newton s Interpolation Formula Our first practical method for determining the interpolating polynomial p(x) applies only in cases where the x-values x 1,x 2,,x n are equally spaced, separated by a common distance h > This means that x k = x 1 + (k 1)h for k = 1,2,,n Define the firstorder differences y k = y k+1 y k, the second-order differences 2 y k = y k+1 y k, etc These differences may be conveniently tabulated by listing columns for the values of k, x k and y k, then listing the differences of the y k s in a new column y k, then the differences of the y k s in a new column 2 y k, then additional columns for 3 y k, 4 y k, etc as far as possible, ie until you run out of values to take differences with Also recall that the factorial of n is n! = n 5

6 Theorem 3 (Newton s Interpolation Formula) If x k = x 1 + (k 1)h for k = 1, 2,, n, then the polynomial p(x) of Theorem 1 is given by p(x) = y 1 + y 1 x x 1 h + + n 1 y 1 (n 1)! + 2 y 1 (x x 1 )(x x 2 ) + 3 y 1 (x x 1 )(x x 2 )(x x 3 ) 2! h 2 3! h 3 (x x 1 )(x x 2 )(x x 3 ) (x x n 1 ) h n 1 For the above example, we obtain the table of values k x k y k y k 2 y k In our case n = 2, h = 1 and the interpolating polynomial is p(x) = 7 + ( 1) x x(x 1) 1 = 7 3x + 2x 2, which agrees with the answer we obtained from our linear system Using Theorem 3, it is possible to deduce the following: Proposition 4 Given data points (x i,y i ) with equally spaced values x i, there exists an interpolating polynomial p(x) of degree < n iff k y j = for all k n As an illustration of Proposition 4, consider the following table of values obtained from the polynomial 7 3x+2x 2 ; note that the differences k y l are zero for k > 2 This should remind you of the fact that the second-order derivative has constant value 4, and all higher order derivatives are zero x k y k y k 2 y k 3 y k 4 y k 5 y k

7 Also note that for the simple task of evaluating p(x) at other integer values of x beyond the original data (ie extrapolating), it is not necessary to know the polynomial p(x) explicitly; we only need to extend the evident pattern of tabulated differences as far as required In the example above, starting with the three rows of the table for x =,1,2, we extend the table as far as desired starting with the rightmost columns (where the pattern is most obvious) and working toward the left This allows us to determine p(x) for x = 2, 1, 3 without having to compute a, b, c and substituting x-values into the resulting polynomial Lagrange s Interpolation Formula We give the corresponding interpolation formula for the more general case when the x k s are not necessarily equally spaced Again, we are looking for the polynomial p(x) of degree < n which passes through the points (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) First we define the polynomials φ j (x) = 1 i n i j x x i x j x i = (x x 1 )(x x 2 ) (x x j 1 )(x x j+1 ) (x x n ) (x j x 1 )(x j x 2 ) (x j x j 1 )(x j x j+1 ) (x j x n ) for j = 1,2,,n Observe that the product has n 1 factors in this case (for all values of i = 1,2,,n excluding j) Also, it is easy to see that { 1, for i = j; φ j (x i ) =, for i j Theorem 5 (Lagrange s Interpolation Formula) The interpolating polynomial p(x) of Theorem 1 is given by p(x) = y 1 φ 1 (x) + y 2 φ 2 (x) + + y n φ n (x) Proof In the sum p(x i ) = n j=1 y jφ j (x i ), all terms are zero except the term for j = i, and we get p(x i ) = y i 1 = y i So the polynomial p(x) = n j=1 y jφ j (x) passes through the n data points (x 1,y 1 ), (x 2,y 2 ),, (x n,y n ) Each φ j (x) is a polynomial in x of degree < n, so n j=1 y jφ j (x) has degree < n So p(x) = n j=1 y jφ j (x) must be the unique polynomial given in Theorem 1 7

8 As an example, we determine the unique polynomial of degree < 3 through the three points (, 5), (1, 2), (3,1) Notice that in this case the x k s are not equally spaced, so that Newton s Interpolation Formula does not apply We compute (x 1)(x 3) φ 1 (x) = ( 1)( 3) = 1 3 (x2 4x + 3), (x )(x 3) φ 2 (x) = (1 )(1 3) = 1 2 (x2 3x), (x )(x 1) φ 3 (x) = (3 )(3 1) = 1 6 (x2 x), p(x) = 5φ 1 (x) 2φ 2 (x) + 1φ 3 (x) = 5 3 (x2 4x + 3) + (x 2 3x) (x2 x) = x 2 + 2x 5 Again, you should check that this polynomial does in fact pass through the three points given HOMEWORK #4 Due Wed March 28 In each of the following, use the field F = Q of rational numbers 1 Let n be a positive integer, and consider the unique polynomial P(x) of degree < n such that P(k) = 2 k for k =,1,2,,n 1 Determine P(n) and prove that your answer is correct (Hint: It is not necessary to explicitly determine the polynomial P(x) Use the pattern evident in a table of differences, and consider the remarks following Proposition 4) 2 Consider the five data points (, 5), (2, 15), (4, 25), (6,13), (8,147) Observe that the x-values are equally spaced (a) What is h? (b) Construct a table of values for k, x k, y k, y k, 2 y k, 3 y k, 4 y k (c) Use Newton s Interpolation Formula to find a polynomial p(x) of degree < 5 passing through the five data points (d) Check that your answer to (c) is correct by evaluating p(x k ) for k = 1,2,3,4,5 (e) What is the degree of p(x)? (f) Explain the relationship between your answers to (b) and (e), using Proposition 4 8

9 3 Use Lagrange s Interpolation Formula to find a polynomial of degree < 4 passing through the four data points ( 1,16), (,19), (1,22), (3, 2) Check that your polynomial is correct by evaluating p(x k ) for k = 1,2,3,4 4 Find the unique polynomial function y = p(x) of degree < 3 passing through the three points ( 3, 3), (, 12), (3, ) (a) by solving the appropriate system of three linear equations in three unknowns; (b) by Newton s Interpolation Formula; (c) by Lagrange s Interpolation Formula Compare your answers in (a), (b) and (c) 9

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 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

### 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

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### 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 +

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### 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

### 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

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 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,

### 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

### Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### 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

### 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

### 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

### Polynomial Invariants

Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,

### LINEAR 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 understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### 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

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### 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

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### 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

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### 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

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### 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

### Math 333 - Practice Exam 2 with Some Solutions

Math 333 - Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### 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

### 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

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### 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

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

### Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

### 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

### MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### 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

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### 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

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

### 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

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### 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

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### 5.3 Determinants and Cramer s Rule

290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

### Using row reduction to calculate the inverse and the determinant of a square matrix

Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### 1 Review of Least Squares Solutions to Overdetermined Systems

cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

### 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

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.