Review of Some Concepts from Linear Algebra: Part 2

Save this PDF as:
Size: px
Start display at page:

Transcription

1 Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, / 22

2 Vector spaces A set of vectors X is called a vector space if for any x, y, z X and any scalars α, β C the following holds 1 x + y X 2 x + y = y + x 3 x + (y + z) = (x + y) + z 4 There exists a unique zero element 0 X, such that x + 0 = x 5 There exists a x X such x + ( x) = 0 6 αx X 7 1 x = x 8 α(x + y) = αx + αy 9 (α + β)x = αx + βx 10 α(βx) = (αβ)x The focus of this course is on the vector spaces R m and C m. Math 566 Linear Algebra Review: Part 2 January 16, / 22

3 Vector spaces A set of vectors X is called a vector space if for any x, y, z X and any scalars α, β C the following holds 1 x + y X 2 x + y = y + x 3 x + (y + z) = (x + y) + z 4 There exists a unique zero element 0 X, such that x + 0 = x 5 There exists a x X such x + ( x) = 0 6 αx X 7 1 x = x 8 α(x + y) = αx + αy 9 (α + β)x = αx + βx 10 α(βx) = (αβ)x The focus of this course is on the vector spaces R m and C m. Math 566 Linear Algebra Review: Part 2 January 16, / 22

4 Vector subspaces A subset of vectors Y of C m is called a subspace if for any two vectors x, y Y and any scalars α, β C, the following holds: αx + βy Y Example: The subset of vectors of R 3 given by t x = 2t, 3t where t R, is a subspace of R 3. Math 566 Linear Algebra Review: Part 2 January 16, / 22

5 Vector subspaces A subset of vectors Y of C m is called a subspace if for any two vectors x, y Y and any scalars α, β C, the following holds: αx + βy Y Example: The subset of vectors of R 3 given by t x = 2t, 3t where t R, is a subspace of R 3. Math 566 Linear Algebra Review: Part 2 January 16, / 22

6 Linear independence A set of vectors {x 1, x 2,..., x n } is linearly independent if α 1 x 1 + α 2 x α n x n = 0 only holds when α 1 = α 2 = = α n = 0. Example: The set of vectors {[ ] [ ]} 1 1, 1 1 is linearly dependent, but the set of vectors {[ ] [ ]} 1 1, 1 1 is not. Math 566 Linear Algebra Review: Part 2 January 16, / 22

7 Linear independence A set of vectors {x 1, x 2,..., x n } is linearly independent if α 1 x 1 + α 2 x α n x n = 0 only holds when α 1 = α 2 = = α n = 0. Example: The set of vectors {[ ] [ ]} 1 1, 1 1 is linearly dependent, but the set of vectors {[ ] [ ]} 1 1, 1 1 is not. Math 566 Linear Algebra Review: Part 2 January 16, / 22

8 Span, basis, dimension The span of a set of vectors is the subspace formed from all linear combinations of the vectors. A basis for a subspace X is a set of linearly independent vectors that span X. The dimension of a subspace X is the number of vectors in any basis for X. The standard basis for R n or C n is given by the {e 1, e 2,, e n }, where e R n and { 0 i j (e i ) j =, i, j = 1, 2,..., n, 1 i = j i.e. e i contains all zeros except for one 1 in the i th entry. Math 566 Linear Algebra Review: Part 2 January 16, / 22

9 Span, basis, dimension The span of a set of vectors is the subspace formed from all linear combinations of the vectors. A basis for a subspace X is a set of linearly independent vectors that span X. The dimension of a subspace X is the number of vectors in any basis for X. The standard basis for R n or C n is given by the {e 1, e 2,, e n }, where e R n and { 0 i j (e i ) j =, i, j = 1, 2,..., n, 1 i = j i.e. e i contains all zeros except for one 1 in the i th entry. Math 566 Linear Algebra Review: Part 2 January 16, / 22

10 Span, basis, dimension The span of a set of vectors is the subspace formed from all linear combinations of the vectors. A basis for a subspace X is a set of linearly independent vectors that span X. The dimension of a subspace X is the number of vectors in any basis for X. The standard basis for R n or C n is given by the {e 1, e 2,, e n }, where e R n and { 0 i j (e i ) j =, i, j = 1, 2,..., n, 1 i = j i.e. e i contains all zeros except for one 1 in the i th entry. Math 566 Linear Algebra Review: Part 2 January 16, / 22

11 Span, basis, dimension The span of a set of vectors is the subspace formed from all linear combinations of the vectors. A basis for a subspace X is a set of linearly independent vectors that span X. The dimension of a subspace X is the number of vectors in any basis for X. The standard basis for R n or C n is given by the {e 1, e 2,, e n }, where e R n and { 0 i j (e i ) j =, i, j = 1, 2,..., n, 1 i = j i.e. e i contains all zeros except for one 1 in the i th entry. Math 566 Linear Algebra Review: Part 2 January 16, / 22

12 Fundamental subspaces of a matrix Let A C m n then the following are the four fundamental subspaces of A: Column space of A (or range of A): the subspace of C m formed by the span of the columns of A. Denoted as C(A). Row space of A (or range of A T ): the subspace of C n formed by the span of the rows of A. Denoted by C(A T ). Null space of A: The subspace of vectors x C n that satisfy Ax = 0. Denoted by N (A). Left null space of A: The subspace of vectors y C m that satisfy y T A = 0. Denoted by N (A T ). Math 566 Linear Algebra Review: Part 2 January 16, / 22

13 Fundamental subspaces of a matrix Let A C m n then the following are the four fundamental subspaces of A: Column space of A (or range of A): the subspace of C m formed by the span of the columns of A. Denoted as C(A). Row space of A (or range of A T ): the subspace of C n formed by the span of the rows of A. Denoted by C(A T ). Null space of A: The subspace of vectors x C n that satisfy Ax = 0. Denoted by N (A). Left null space of A: The subspace of vectors y C m that satisfy y T A = 0. Denoted by N (A T ). Math 566 Linear Algebra Review: Part 2 January 16, / 22

14 Fundamental subspaces of a matrix Let A C m n then the following are the four fundamental subspaces of A: Column space of A (or range of A): the subspace of C m formed by the span of the columns of A. Denoted as C(A). Row space of A (or range of A T ): the subspace of C n formed by the span of the rows of A. Denoted by C(A T ). Null space of A: The subspace of vectors x C n that satisfy Ax = 0. Denoted by N (A). Left null space of A: The subspace of vectors y C m that satisfy y T A = 0. Denoted by N (A T ). Math 566 Linear Algebra Review: Part 2 January 16, / 22

15 Fundamental subspaces of a matrix Let A C m n then the following are the four fundamental subspaces of A: Column space of A (or range of A): the subspace of C m formed by the span of the columns of A. Denoted as C(A). Row space of A (or range of A T ): the subspace of C n formed by the span of the rows of A. Denoted by C(A T ). Null space of A: The subspace of vectors x C n that satisfy Ax = 0. Denoted by N (A). Left null space of A: The subspace of vectors y C m that satisfy y T A = 0. Denoted by N (A T ). Math 566 Linear Algebra Review: Part 2 January 16, / 22

16 Fundamental theorem of linear algebra Part 1: Let A C m n then the column space C(A) and row space C(A T ) have dimension r min(m, n) and the null space N (A) and N (A T ) have dimension n r and m r, respectively. The dimension r of C(A) and C(A T ) is called the rank of A. If m n and r = n then A is said to be of full column rank. If n m and r = m then A is said to be of full row rank. If r < min(m, n) then A is said to be rank deficient. Math 566 Linear Algebra Review: Part 2 January 16, / 22

17 Fundamental theorem of linear algebra Part 1: Let A C m n then the column space C(A) and row space C(A T ) have dimension r min(m, n) and the null space N (A) and N (A T ) have dimension n r and m r, respectively. The dimension r of C(A) and C(A T ) is called the rank of A. If m n and r = n then A is said to be of full column rank. If n m and r = m then A is said to be of full row rank. If r < min(m, n) then A is said to be rank deficient. Math 566 Linear Algebra Review: Part 2 January 16, / 22

18 Fundamental theorem of linear algebra Part 1: Let A C m n then the column space C(A) and row space C(A T ) have dimension r min(m, n) and the null space N (A) and N (A T ) have dimension n r and m r, respectively. The dimension r of C(A) and C(A T ) is called the rank of A. If m n and r = n then A is said to be of full column rank. If n m and r = m then A is said to be of full row rank. If r < min(m, n) then A is said to be rank deficient. Math 566 Linear Algebra Review: Part 2 January 16, / 22

19 Fundamental theorem of linear algebra Part 2: Let A C m n then the column space C(A) is the orthogonal complement of the left null space N (A T ) and the row space C(A T ) is the orthogonal complement of the null space N (A). This means: For any y N (A T ), y T z = 0 for all z C(A). For any x N (A), x T w = 0 for all w C(A T ). C m = C(A) N (A T ) C n = C(A T ) N (A) Math 566 Linear Algebra Review: Part 2 January 16, / 22

20 Fundamental theorem of linear algebra Part 2: Let A C m n then the column space C(A) is the orthogonal complement of the left null space N (A T ) and the row space C(A T ) is the orthogonal complement of the null space N (A). This means: For any y N (A T ), y T z = 0 for all z C(A). For any x N (A), x T w = 0 for all w C(A T ). C m = C(A) N (A T ) C n = C(A T ) N (A) Math 566 Linear Algebra Review: Part 2 January 16, / 22

21 Fundamental theorem of linear algebra Part 2: Let A C m n then the column space C(A) is the orthogonal complement of the left null space N (A T ) and the row space C(A T ) is the orthogonal complement of the null space N (A). This means: For any y N (A T ), y T z = 0 for all z C(A). For any x N (A), x T w = 0 for all w C(A T ). C m = C(A) N (A T ) C n = C(A T ) N (A) Math 566 Linear Algebra Review: Part 2 January 16, / 22

22 Fundamental theorem of linear algebra Part 3: This is about the Singular Value Decomposition (SVD) and we will cover this later. Math 566 Linear Algebra Review: Part 2 January 16, / 22

23 Invertibility of a matrix Theorem: For a square matrix A C n n the following are equivalent: A 1 exists rank(a) = n Columns of A form a basis for C n, i.e. C(A) = C n Rows of A form a basis for C n, i.e. C(A T ) = C n N (A) = {0} Math 566 Linear Algebra Review: Part 2 January 16, / 22

24 Eigenvalues and eigenvectors Let A C n n then we call λ C an eigenvalue of A if there exists a vector x C n with x 0 such that Ax = λx. x is called an eigenvector corresponding to the eigenvalue λ. Math 566 Linear Algebra Review: Part 2 January 16, / 22

25 Eigenvalues and eigenvectors The scalar λ C is an eigenvalue of A C n n if and only if det(a Iλ) = 0, where I is the identity matrix. p n (λ) = det(a Iλ) is a polynomial in λ of degree n. p n (λ) is called the characteristic polynomial of A. We never use p n (λ) to numerically compute the eigenvalues of A. We will discuss the correct way to compute the eigenvalues. Math 566 Linear Algebra Review: Part 2 January 16, / 22

26 Eigenvalues and eigenvectors The scalar λ C is an eigenvalue of A C n n if and only if det(a Iλ) = 0, where I is the identity matrix. p n (λ) = det(a Iλ) is a polynomial in λ of degree n. p n (λ) is called the characteristic polynomial of A. We never use p n (λ) to numerically compute the eigenvalues of A. We will discuss the correct way to compute the eigenvalues. Math 566 Linear Algebra Review: Part 2 January 16, / 22

27 Eigenvalues and eigenvectors The scalar λ C is an eigenvalue of A C n n if and only if det(a Iλ) = 0, where I is the identity matrix. p n (λ) = det(a Iλ) is a polynomial in λ of degree n. p n (λ) is called the characteristic polynomial of A. We never use p n (λ) to numerically compute the eigenvalues of A. We will discuss the correct way to compute the eigenvalues. Math 566 Linear Algebra Review: Part 2 January 16, / 22

28 Eigenvalues and eigenvectors The scalar λ C is an eigenvalue of A C n n if and only if det(a Iλ) = 0, where I is the identity matrix. p n (λ) = det(a Iλ) is a polynomial in λ of degree n. p n (λ) is called the characteristic polynomial of A. We never use p n (λ) to numerically compute the eigenvalues of A. We will discuss the correct way to compute the eigenvalues. Math 566 Linear Algebra Review: Part 2 January 16, / 22

29 Vector norms A vector norm is a scalar quantity that reflects the size of a vector x. The norm of a vector x is denoted as x. There are many ways to define the size of a vector. If x C n, the three most popular are one-norm : x 1 = n x k, k=1 two-norm : x 2 = n x k 2, -norm : k=1 x = max 1 k n x k Math 566 Linear Algebra Review: Part 2 January 16, / 22

30 Vector norms However a vector norm is defined, it must satisfy the following three properties to be called a norm: 1 x 0 and x = 0 if and only if x = 0 (i.e. x contains all zeros as its entries). 2 αx = α x, for any constant α. 3 x + y x + y, where y C n. This is called the triangle inequality. Math 566 Linear Algebra Review: Part 2 January 16, / 22

31 Unit vectors A vector x is called a unit vector if its norm is one, i.e. x = 1. Unit vectors will be different depending on the norm applied. Below are several unit vectors in the one, two, and norms for x R 2. 1 Unit vectors with respect to the one norm 1 Unit vectors with respect to the two norm 1 Unit vectors with respect to the norm x 2 0 x 2 0 x x x x 1 (a) One-norm (b) Two-norm (c) -norm Math 566 Linear Algebra Review: Part 2 January 16, / 22

32 Matrix norms A matrix norm is a scalar quantity that reflects the size of a matrix A C m n. The norm of A is denoted as A. Any matrix norm must satisfy the following four properties: 1 A 0 and A = 0 if and only if A = 0 (i.e. A contains all zeros as its entries). 2 αa = α A, for any constant α. 3 A + B A + B, where B C m n. 4 AB A B, where B C n p. This is called the submultiplicative inequality. Math 566 Linear Algebra Review: Part 2 January 16, / 22

33 Matrix norms Each vector norm induces a matrix norm according to the following definition: Ax p A p = max x p 0 x p where x C n and p = 1, 2,.... = max x p=1 Ax p, Induced norms describe how the matrix stretches unit vectors with respect to that norm. Math 566 Linear Algebra Review: Part 2 January 16, / 22

34 Induced matrix norms Two popular and easy to define induced matrix norms are One-norm : -norm : A 1 = max 1 k n A = max m a jk, j=1 1 j m k=1 n a jk. The one-norm corresponds to the maximum of the one norm of every column. The -norm corresponds to the maximum of the one norm of every row. Math 566 Linear Algebra Review: Part 2 January 16, / 22

35 Induced matrix norms The two-norm is also a popular induced matrix norm, but is not easily derived. Definition: Let B C n n then the spectral radius of B is ρ(b) = max 1 j n λ j, where λ j are the n eigenvalues of B. Theorem: Let A C m n then A 2 = ρ(a A). Math 566 Linear Algebra Review: Part 2 January 16, / 22

36 Geometric illustration of the two-norm [ ] 4 1 Let A = and consider applying A to all vectors x such that 1 3 x 2 = (a) Unit vectors w.r.t the two-norm (b) Transformation of unit vectors after applying A The semi-major axis of the ellipse on the right (marked in black) is the two-norm of A. Math 566 Linear Algebra Review: Part 2 January 16, / 22

37 Non-induced matrix norms The most popular matrix norm that is not an induced norm is the Frobenius norm: m n A F = a jk 2. j=1 k=1 Math 566 Linear Algebra Review: Part 2 January 16, / 22

38 Important results on matrix norms The following are some useful inequalities involving matrix norms. Here A R m n : ρ(a) A for any matrix norm Ax A x 1 m A 1 A 2 n A 1 1 n A A 2 m A A 2 A F n A 2 A 2 A 1 A Math 566 Linear Algebra Review: Part 2 January 16, / 22

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

Computational math: Assignment 1

Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange

Math 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:

Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your

Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

Elementary 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

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

UNIT 6: The singular value decomposition.

UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T

ELE/MCE 503 Linear Algebra Facts Fall 2018

ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.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

Linear Algebra Formulas. Ben Lee

Linear Algebra Formulas Ben Lee January 27, 2016 Definitions and Terms Diagonal: Diagonal of matrix A is a collection of entries A ij where i = j. Diagonal Matrix: A matrix (usually square), where entries

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. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

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

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 +

Applied Linear Algebra in Geoscience Using MATLAB

Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

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

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

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

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

MAT 610: Numerical Linear Algebra. James V. Lambers

MAT 610: Numerical Linear Algebra James V Lambers January 16, 2017 2 Contents 1 Matrix Multiplication Problems 7 11 Introduction 7 111 Systems of Linear Equations 7 112 The Eigenvalue Problem 8 12 Basic

Section 3.9. Matrix Norm

3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix

Linear 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

A Review of Linear Algebra

A Review of Linear Algebra Mohammad Emtiyaz Khan CS,UBC A Review of Linear Algebra p.1/13 Basics Column vector x R n, Row vector x T, Matrix A R m n. Matrix Multiplication, (m n)(n k) m k, AB BA. Transpose

Linear Algebra Review

January 29, 2013 Table of contents Metrics Metric Given a space X, then d : X X R + 0 and z in X if: d(x, y) = 0 is equivalent to x = y d(x, y) = d(y, x) d(x, y) d(x, z) + d(z, y) is a metric is for all

MAT Linear Algebra Collection of sample exams

MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system

Algebra C Numerical Linear Algebra Sample Exam Problems

Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric

I. Multiple Choice Questions (Answer any eight)

Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY

Basic Elements of Linear Algebra

A Basic Review of Linear Algebra Nick West nickwest@stanfordedu September 16, 2010 Part I Basic Elements of Linear Algebra Although the subject of linear algebra is much broader than just vectors and matrices,

Math 265 Linear Algebra Sample Spring 2002., rref (A) =

Math 265 Linear Algebra Sample Spring 22. It is given that A = rref (A T )= 2 3 5 3 2 6, rref (A) = 2 3 and (a) Find the rank of A. (b) Find the nullityof A. (c) Find a basis for the column space of A.

CS 246 Review of Linear Algebra 01/17/19

1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

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

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

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.

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian

FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.

1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III

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

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.

Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where

MA 265 FINAL EXAM Fall 2012

MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

SUMMARY OF MATH 1600

SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You

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

Jim Lambers MAT 610 Summer Session Lecture 1 Notes

Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra

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,

There are six more problems on the next two pages

Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with

Linear Algebra. Session 12

Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)

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

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

LinGloss. A glossary of linear algebra

LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

MATH 304 Linear Algebra Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vector spaces.

MATH 304 Linear Algebra Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vector spaces. Orthogonality Definition 1. Vectors x,y R n are said to be orthogonal (denoted x y)

Linear Algebra Review. Vectors

Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

Linear Algebra Review: Linear Independence. IE418 Integer Programming. Linear Algebra Review: Subspaces. Linear Algebra Review: Affine Independence

Linear Algebra Review: Linear Independence IE418: Integer Programming Department of Industrial and Systems Engineering Lehigh University 21st March 2005 A finite collection of vectors x 1,..., x k R n

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

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

Linear Algebra Practice Problems

Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,

10-701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang)

10-701/15-781 Recitation : Linear Algebra Review (based on notes written by Jing Xiang) Manojit Nandi February 1, 2014 Outline Linear Algebra General Properties Matrix Operations Inner Products and Orthogonal

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

Lecture 1: Review of linear algebra

Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations

Mathematical foundations - linear algebra

Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =

30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can

Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.

Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has

Foundations of Matrix Analysis

1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the

Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

Lecture: Linear algebra. 4. Solutions of linear equation systems The fundamental theorem of linear algebra

Lecture: Linear algebra. 1. Subspaces. 2. Orthogonal complement. 3. The four fundamental subspaces 4. Solutions of linear equation systems The fundamental theorem of linear algebra 5. Determining the fundamental

Problem # Max points possible Actual score Total 120

FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to

Review Notes for Linear Algebra True or False Last Updated: February 22, 2010

Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n

Linear System Theory

Linear System Theory Wonhee Kim Lecture 4 Apr. 4, 2018 1 / 40 Recap Vector space, linear space, linear vector space Subspace Linearly independence and dependence Dimension, Basis, Change of Basis 2 / 40

Lecture 10 - Eigenvalues problem

Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems

AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES

AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim

Lecture 7: Positive Semidefinite Matrices

Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m

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

Inner products and Norms. Inner product of 2 vectors. Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y x n y n in R n

Inner products and Norms Inner product of 2 vectors Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y 2 + + x n y n in R n Notation: (x, y) or y T x For complex vectors (x, y) = x 1 ȳ 1 + x 2

Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

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

2018 Fall 2210Q Section 013 Midterm Exam II Solution

08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

Math Final December 2006 C. Robinson

Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the

MATH 1553 PRACTICE FINAL EXAMINATION

MATH 553 PRACTICE FINAL EXAMINATION Name Section 2 3 4 5 6 7 8 9 0 Total Please read all instructions carefully before beginning. The final exam is cumulative, covering all sections and topics on the master

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

CS 143 Linear Algebra Review

CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see

Numerical Linear Algebra

Numerical Linear Algebra The two principal problems in linear algebra are: Linear system Given an n n matrix A and an n-vector b, determine x IR n such that A x = b Eigenvalue problem Given an n n matrix

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O

Further 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:

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

Introduction to Numerical Linear Algebra II

Introduction to Numerical Linear Algebra II Petros Drineas These slides were prepared by Ilse Ipsen for the 2015 Gene Golub SIAM Summer School on RandNLA 1 / 49 Overview We will cover this material in

3 (Maths) Linear Algebra

3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra

Linear Algebra- Final Exam Review

Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

1 9/5 Matrices, vectors, and their applications

1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric

Final Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015

Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,