SOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4.

Save this PDF as:

Size: px
Start display at page:

Download "SOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4."

Transcription

1 SOLUTIONS: ASSIGNMENT 9 66 Use Gaussian elimination to find the determinant of the matrix det = det = det = 1 ( ) 3 6 = 36 = det = det Consider a 4 4 matrix A with rows v 1, v, v 3, v 4 If det(a) = 8, find v 4 det v v 3 v 1 This matrix is obtained from A by a row swap, so its determinant is det(a) = Using the same row vectors as in 61, 6 v 1 + v 4 det v v 3 3 v 1 + v 4 The rows of this matrix are linearly dependent, because (3 v 1 + v 4 ) = 6 v 1 + v 4 A matrix whose rows are linearly dependent must have determinant Find the determinant of the linear transformation T (f(t)) = f(3t ) from P to P A basis for P is {1, t, t } Applying T to each basis element, we have T (1) = 1, T (t) = 3t, and T (t ) = (3t ) = 9t 1t + 4 The matrix of T with respect to this basis is , which has a determinant of 7 1

2 6 Find the determinant of the linear transformation T (f(t)) = f( t) from P n to P n A basis for P n is {t k : 0 k } If k is odd, then T (t k ) = t k, and if k is even then T (t k ) = t k Therefore the determinant of T is ( 1) m where m is the number of odd numbers between 0 and n, inclusive Let N be the remainder when n is divided by 4 A quick check shows that m is even if N is 0 or 3, while m is odd if N is 1 or The n(n + 1) quantity is also even if N is 0 or 3 (because then either n or n+1 is divisible by 4) but is even if N is 1 or Thus the determinant n(n + 1) of T is ( 1) 630 Consider two distinct numbers a and b We define the function f(t) = det a b t a b t (a) Show that f(t) is a quadratic function What is the coefficient of t? Expanding down the third column, we have f(t) = D 1 D t + D 3 t, where the D i are determinants of matrices that contain no factor of t Since D 3 = b a, f(t) is a quadratic function with leading coefficient b a (b) Explain why f(a) = f(b) = 0 Conclude that f(t) = k(t a)(t b), for some constant k Find k, using your work in part (a) If t = a, the first and third columns of the matrix are the same, so it has determinant 0 Likewise, if t = b, the second and third columns of the matrix are the same This shows that f(a) = f(b) = 0 It follows that f(t) has the factors t a and t b; since f(t) is quadratic, it can have no other non-constant factors, so f(t) = k(t a)(t b) for some constant k This constant is equal to the leading coefficient of f(t) which is b a by part (a) (c) For which values of t is the matrix invertible? A matrix is invertible if and only if its determinant is nonzero In part (b) we found that f(t) = (b a)(t a)(t b), which is nonzero for all values of t except a and b

3 631 Vandermonde determinants: Consider distinct scalars a 0, a 1,, a n We define the (n + 1) (n + 1) matrix A = Vandermonde showed that a 0 a 1 a n a 0 a 1 a n a n 0 a n 1 a n n det(a) = i>j(a i a j ), the product of all differences a i a j where i exceeds j (a) Verify this formula in the case of n = 1 [ ] 1 1 If n = 1 then A = and det(a) = a a 0 a 1 a 0 1 (b) Suppose the Vandermonde formula holds for n 1 You are asked to demonstrate it for n Consider the function a 0 a 1 a n 1 t f(t) = det a 0 a 1 a n 1 t a n 0 a n 1 a n n 1 t n Explain why f(t) is a polynomial of nth degree Find the coefficient k of t n using Vandermonde s formula for n 1 Explain why f(a 0 ) = f(a 1 ) = = f(a n 1 ) = 0 Conclude that f(t) = k(t a 0 )(t a 1 ) (t a n 1 ) for the scalar k you found above Substitute t = a n to demonstrate Vandermonde s formula Expanding down the rightmost column of the matrix, we have f(t) = ±D 0 D 1 t ± + D n t n 3

4 where the D i are determinants of n n matrices that contain no factor of t Since D n is the Vandermonde determinant of the n n matrix with coefficients a 0 through a n 1, we conclude that f(t) is an nth degree polynomial with leading coefficient k = n>i>j (a i a j ) If t = a 0, then f(t) = 0 because the leftmost and rightmost columns of the matrix are equal Applying the same reasoning to a 1 through a n 1, we can see that f(a 0 ) = f(a 1 ) = = f(a n 1 ) = 0 Since the n values a i for 0 i n are all distinct, and f(t) is an nth degree polynomial, it must be the case that f(t) = k(t a 0 )(t a 1 ) (t a n 1 ) for the value of k found above As stated, substituting t = a n into this equation demonstrates Vandermonde s formula for n 63 Find the area of the triangle defined by [ 3 7 ] and [ 8 The area of the triangle formed by two vectors is half the area of the parallelogram formed by the same two vectors Thus, the area is [ ] det = 1 (56 6 ) = Consider the area A of a triangle with vertices Express A in terms of D = det a 1 b 1 c 1 a b c [ a1 a ] ] [ b1, b ] [ c1, c The answer is A = D On one hand, note that A is the same as [ ] [ ] [ ] 0 b1 a the area of the triangle with vertices, 1 c1 a, 1 0 b a c a This area is half the absolute value of the quantity [ ] b1 a det 1 c 1 a 1 = (b b a c a 1 a 1 )(c a ) (c 1 a 1 )(b a ) 4 ]

5 On the other hand, D = (a 1 b b 1 a ) + (c 1 a a 1 c ) + (b 1 c c 1 b ) = (b 1 a 1 )c a 1 c + (a 1 c 1 )b + a 1 b = (b 1 a 1 )(c a ) (c 1 a 1 )(b a ) where 0 = a 1 a a 1 a was added to the equation in the last step 6310 Consider an n n matrix A = [ v 1 v v n ] What is the relationship between the product v 1 v v n and det(a)? When is det(a) = ṽ 1 ṽ ṽ n? Using a fact proven in the textbook (Fact 634), det(a) = ṽ 1 ṽ ṽ n = ṽ 1 ṽ ṽ n with equality if and only if v i = v i for all i; that is, if the v i form an orthogonal basis for R n 636 Consider an n n matrix A with integer entries such that det (A) = 1 Are the entries of A 1 necessarily integers? Explain Yes, A 1 must have integer entries Since det(a) = 1, A 1 = adj(a) But adj(a) has integer entries because A does 71 Let A be an invertible n n matrix and v an eigenvector of A with associated eigenvalue λ Is v an eigenvector of A 1? If so, what is the eigenvalue? We can see that v is an eigenvector of A 1 when we multiply the equation A v = λv by A 1, as follows: A 1 v = 1 λ A 1 (λ v) = 1 λ A 1 (A v) = 1 λ v, so the eigenvalue is 1 λ 713 Let A be an invertible n n matrix and v an eigenvector of A with associated eigenvalue λ Is v an eigenvector of A + I n? If so, what is the eigenvalue? Yes, because (A + I n )( v) = A v + I n v = λ v + v = (λ + ) v, so the eigenvalue is λ + 5

6 715 If a vector v is an eigenvector of both A and B, is v necessarily an eigenvector of A + B? Yes If the eigenvalues for A and B are α and β, respectively, then (A + B)( v) = A v + B v = α v + β v = (α + β) v, so the eigenvalue for A + B is α + β 716 If a vector v is an eigenvector of both A and B, is v necessarily an eigenvector of AB? Yes If the eigenvalues for A and B are α and β, respectively, then (AB)( v) = A(B v) = A(β v) = β(a v) = βα v = αβ v, so the eigenvalue for AB is αβ 7116 Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformation that represents rotation by 180 degrees in R Find a basis of eigenvectors If T is the name of this transformation, then T v = v for every v in R Every vector in R is an eigenvector of T with eigenvalue 1 The standard basis for R is a basis of eigenvectors, for example 7134 Suppose v is an eigenvector of the n n matrix A, with eigenvalue 4 Explain why v is an eigenvector of A + A + 3I n What is the associated eigenvalue? By repeatedly using the equation A v = 4 v, we find that (A + A + 3I n )( v) = A v + A v + 3I n v = 16 v + 4 v + 3 v = 7 v, so the associated eigenvalue is 7 [ ] [ ] Find a matrix A such that and 1 A, with eigenvalues 5 and 10, respectively [ ] a b Write A = We obtain the equations c d are eigenvectors of 3a + b = 15 3c + d = 5 a + b = 10 c + d = 0 The solution to this system is a = 4, b = 3, c =, d = 11 6

7 7144 For m n, find the dimension of the space V of all n n matrices A for which all the vectors e 1,, e n are eigenvectors The dimension of V is m + n(n m) Suppose A is a matrix in V If the corresponding eigenvalues are λ 1,, λ m then the first m columns of A are λ 1 e 1,, λ m e m Each of these eigenvalues represents one free variable The remaining n m columns can be arbitrary vectors in R n, for a total of n(n m) free variables The number of free variables for the entire matrix is thus m + n(n m) 7

235 Final exam review questions

5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

Math 215 HW #9 Solutions

Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

and let s calculate the image of some vectors under the transformation T.

Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra A linear (algebraic) 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 and b are real numbers. 1

Chapter SSM: Linear Algebra Section Fails to be invertible; since det = 6 6 = Invertible; since det = = 2.

SSM: Linear Algebra Section 61 61 Chapter 6 1 2 1 Fails to be invertible; since det = 6 6 = 0 3 6 3 5 3 Invertible; since det = 33 35 = 2 7 11 5 Invertible; since det 2 5 7 0 11 7 = 2 11 5 + 0 + 0 0 0

Diagonalization. Hung-yi Lee

Diagonalization Hung-yi Lee Review If Av = λv (v is a vector, λ is a scalar) v is an eigenvector of A excluding zero vector λ is an eigenvalue of A that corresponds to v Eigenvectors corresponding to λ

Review of Linear Algebra

Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

Linear Algebra: Matrix Eigenvalue Problems

CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.

The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section

Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that

Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that det(a) = 2, det(b) = 2, det(c) = 1, det(d) = 4. 2 (a) Compute det(ad)+det((b

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

This MUST hold matrix multiplication satisfies the distributive property.

The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients

1. 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

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

Eigenvalues and Eigenvectors

Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying

MATH 221, Spring Homework 10 Solutions

MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the

Linear Algebra - Part II

Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T

Matrix Operations: Determinant

Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant

MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial.

MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial. Geometric properties of determinants 2 2 determinants and plane geometry

Math Matrix Algebra

Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,

Eigenvalues and Eigenvectors

5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

Math 315: Linear Algebra Solutions to Assignment 7

Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

1 Last time: linear systems and row operations

1 Last time: linear systems and row operations Here s what we did last time: a system of linear equations or linear system is a list of equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)-matrix Consider the following row operations on A (1) Swap the positions any

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

1. 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

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

Solutions Problem Set 8 Math 240, Fall

Solutions Problem Set 8 Math 240, Fall 2012 5.6 T/F.2. True. If A is upper or lower diagonal, to make det(a λi) 0, we need product of the main diagonal elements of A λi to be 0, which means λ is one of

Introduction to Matrices

POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder

Linear Algebra Practice Problems

Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a

Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

2. Every linear system with the same number of equations as unknowns has a unique solution.

1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

Conceptual Questions for Review

Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues.

Similar Matrices and Diagonalization Page 1 Theorem If A and B are n n matrices, which are similar, then they have the same characteristic equation and hence the same eigenvalues. Proof Let A and B be

Chapter 3 Transformations

Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

Chapter 3 Transformations

Chapter 3 Transformations An Introduction to Optimization Spring, 2015 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

MATH 1553 PRACTICE MIDTERM 3 (VERSION B)

MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

Lecture 11: Eigenvalues and Eigenvectors

Lecture : Eigenvalues and Eigenvectors De nition.. Let A be a square matrix (or linear transformation). A number λ is called an eigenvalue of A if there exists a non-zero vector u such that A u λ u. ()

LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU

LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:

MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor

Econ Slides from Lecture 7

Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for

BASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x

BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,

Eigenvalues and Eigenvectors

5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

Computationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:

Diagonalization We have seen that diagonal and triangular matrices are much easier to work with than are most matrices For example, determinants and eigenvalues are easy to compute, and multiplication

Section 8.2 : Homogeneous Linear Systems

Section 8.2 : Homogeneous Linear Systems Review: Eigenvalues and Eigenvectors Let A be an n n matrix with constant real components a ij. An eigenvector of A is a nonzero n 1 column vector v such that Av

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12

24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

4. Linear transformations as a vector space 17

4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation

18.S34 linear algebra problems (2007)

18.S34 linear algebra problems (2007) Useful ideas for evaluating determinants 1. Row reduction, expanding by minors, or combinations thereof; sometimes these are useful in combination with an induction

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See

MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

Computational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science

Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (non-zero) with corresponding

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij

Matrices and Linear Algebra

Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

1. Select the unique answer (choice) for each problem. Write only the answer.

MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

k is a product of elementary matrices.

Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation

Math 1553 Worksheet 5.3, 5.5

Math Worksheet, Answer yes / no / maybe In each case, A is a matrix whose entries are real a) If A is a matrix with characteristic polynomial λ(λ ), then the - eigenspace is -dimensional b) If A is an

MATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005

MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all

88 CHAPTER 3. SYMMETRIES

88 CHAPTER 3 SYMMETRIES 31 Linear Algebra Start with a field F (this will be the field of scalars) Definition: A vector space over F is a set V with a vector addition and scalar multiplication ( scalars

Final Exam Practice Problems Answers Math 24 Winter 2012

Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

MATRICES. a m,1 a m,n A =

MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

Determining a span. λ + µ + ν = x 2λ + 2µ 10ν = y λ + 3µ 9ν = z.

Determining a span Set V = R 3 and v 1 = (1, 2, 1), v 2 := (1, 2, 3), v 3 := (1 10, 9). We want to determine the span of these vectors. In other words, given (x, y, z) R 3, when is (x, y, z) span(v 1,

NOTES 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

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM () We find a least squares solution to ( ) ( ) A x = y or 0 0 a b = c 4 0 0. 0 The normal equation is A T A x = A T y = y or 5 0 0 0 0 0 a b = 5 9. 0 0 4 7

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that

SPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS

SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the

Announcements Wednesday, November 7

Announcements Wednesday, November 7 The third midterm is on Friday, November 6 That is one week from this Friday The exam covers 45, 5, 52 53, 6, 62, 64, 65 (through today s material) WeBWorK 6, 62 are

4. Determinants.

4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

A = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is,

65 Diagonalizable Matrices It is useful to introduce few more concepts, that are common in the literature Definition 65 The characteristic polynomial of an n n matrix A is the function p(λ) det(a λi) Example

TBP MATH33A Review Sheet. November 24, 2018

TBP MATH33A Review Sheet November 24, 2018 General Transformation Matrices: Function Scaling by k Orthogonal projection onto line L Implementation If we want to scale I 2 by k, we use the following: [

Definition (T -invariant subspace) Example. Example

Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

Proofs for Quizzes. Proof. Suppose T is a linear transformation, and let A be a matrix such that T (x) = Ax for all x R m. Then

Proofs for Quizzes 1 Linear Equations 2 Linear Transformations Theorem 1 (2.1.3, linearity criterion). A function T : R m R n is a linear transformation if and only if a) T (v + w) = T (v) + T (w), for

Chapter 5. Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the

Solving a system by back-substitution, checking consistency of a system (no rows of the form

MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary

Inverses. Stephen Boyd. EE103 Stanford University. October 28, 2017

Inverses Stephen Boyd EE103 Stanford University October 28, 2017 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number

Practice problems for Exam 3 A =

Practice problems for Exam 3. Let A = 2 (a) Determine whether A is diagonalizable. If so, find a matrix S such that S AS is diagonal. If not, explain why not. (b) What are the eigenvalues of A? Is A diagonalizable?

Linear Systems and Matrices

Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) 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.......

Eigenvalues by row operations

Eigenvalues by row operations Barton L. Willis Department of Mathematics University of Nebraska at Kearney Kearney, Nebraska 68849 May, 5 Introduction There is a wonderful article, Down with Determinants!,

Cayley-Hamilton Theorem

Cayley-Hamilton Theorem Massoud Malek 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 Let A be an n n matrix Although det (λ I n A

Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates

Online Exercises for Linear Algebra XM511

This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2