Matrix invertibility. Rank-Nullity Theorem: For any n-column matrix A, nullity A +ranka = n

Size: px
Start display at page:

Download "Matrix invertibility. Rank-Nullity Theorem: For any n-column matrix A, nullity A +ranka = n"

Transcription

1 Matrix invertibility Rank-Nullity Theorem: For any n-column matrix A, nullity A +ranka = n Corollary: Let A be an R C matrix. Then A is invertible if and only if R = C and the columns of A are linearly independent. Proof: Let F be the field. Define f : F C! F R by f (x) =Ax. Then A is an invertible matrix if and only if f is an invertible function. The function f is invertible i dim Ker f =0anddimF C =dimf R i nullity A =0and C = R. nullity A =0 i dim Null A =0 i Null A = {0} i the only vector x such that Ax = 0 is x = 0 i the columns of A are linearly independent. QED

2 Matrix invertibility examples apple is not square so cannot be invertible. apple is square and its columns are linearly independent so it is invertible. 5 is square but columns not linearly independent so it is not invertible.

3 Transpose of invertible matrix is invertible Theorem: The transpose of an invertible matrix is invertible. a 1 v 1 v n A = 5 = 6. a n 5 A T = a 1 a n 5 Proof: Suppose A is invertible. Then A is square and its columns are linearly independent. Let n be the number of columns. Then rank A = n. Because A is square, it has n rows. By Rank Theorem, rows are linearly independent. Columns of transpose A T are rows of A, so columns of A T are linearly independent. Since A T is square and columns are linearly independent, A T is invertible. QED

4 More matrix invertibility Earlier we proved: If A has an inverse A 1 then AA 1 is identity matrix Converse: If BA is identity matrix then A and B are inverses? Not always true. Theorem: Suppose A and B are square matrices such that BA is an identity matrix 1. ThenA and B are inverses of each other. Proof: To show that A is invertible, need to show its columns are linearly independent. Let u be any vector such that Au = 0. Then B(Au) =B0 = 0. On the other hand, (BA)u = 1u = u, so u = 0. This shows A has an inverse A 1. Now must show B = A 1. We know AA 1 = 1. BA = 1 (BA)A 1 = 1A 1 by multiplying on the right by A 1 (BA)A 1 = A 1 B(AA 1 ) = A 1 B 1 = A 1 B = A 1 by associativity of matrix-matrix mult QED

5 Representations of vector spaces Two important ways to represent a vector space: As the solution set of homogeneous linear system a 1 x =0,...,a m x =0 Equivalently, Null 6 a 1. a m 5 As Span {b 1,...,b k } Equivalently, Col 6 b 1 b k 5

6 Conversions between the two representations {[x, y, z] : [, 1, 1] [x, y, z] =0, [0, 1, 1] [x, y, z] =0} Span {[1,, ]} Span {[, 1, 1], [0, 1, 1]} {[x, y, z] : [1,, ] [x, y, z] =0}

7 Conversions for a ne spaces? I From representation as solution set of linear system to representation as a ne hull I From representation as a ne hull to representation as solution set of linear system

8 Conversions for a ne spaces? From representation as solution set of linear system to representation as a ne hull I input: linear system Ax = b I output: vectors whose a ne hull is the solution set of the linear system. apple Let u be one solution to the linear system. u =[ 0.5, 0.5, 0] apple x 1 1 Consider the corresponding homogeneous system Ax = 0. y 1 z Its solution set, the null space of A, isavectorspacev. Let b 1,...,b k be generators for V. b 1 =[,, ] Then the solution set of the original linear system is the a ne hull of u, b 1 + u, b + u,...,b k + u. [ 0.5,.5, 0] and [ 0.5,.5, 0] + [,, ] x y z 5 = 5 = apple 1 apple 0 0

9 From representation as solution set to representation as a One solution to equation Null space of apple apple is Span {b 1 }: x y z 5 = apple 1 is u =[ 0.5, 0.5, 0] ne hull Solution set of equation is u +Span{b 1 }, i.e. the a ne hull of u and u + b 1 b 1 u+b 1 u

10 Representations of vector spaces Two important ways to represent a vector space: As the solution set of homogeneous linear system a 1 x =0,...,a m x =0 Equivalently, Null 6 a 1. a m 5 As Span {b 1,...,b k } Equivalently, Col 6 b 1 b k 5

11 Representations of vector spaces Two important ways to represent a vector space: As the solution set of homogeneous linear system a 1 x =0,...,a m x =0 Equivalently, Null 6 a 1. a m 5 As Span {b 1,...,b k } Equivalently, Col 6 b 1 b k 5 How to transform between these two representations? Problem 1 (From left to right): I input: homogeneous linear system a 1 x =0,...,a m x = 0, I output: basis b 1,...,b k for solution set Problem (From right to left): I input: independent vectors b 1,...,b k, I output: homogeneous linear system a x = 0,...,a x = 0 whose solution set equals

12 Reformulating Problem 1 I input: homogeneous linear system a 1 x =0,...,a m x = 0, I output: basis b 1,...,b k for solution set Let s express this in the language of matrices: I input: matrix A = 6 I output: matrix B = a 1. a m 5 b k b 1 5 such that Col B =NullA Can require the rows of the input matrix A to be linearly independent. (Discarding a superfluous row does not change the null space of A.)

13 Reformulating Problem 1 I input: homogeneous linear system a 1 x =0,...,a m x = 0, I output: basis b 1,...,b k for solution set Let s express this in the language of matrices: I input: matrix A = 6 I output: matrix B = a 1. a m 5 b k b 1 5 with independent columns such that Col B =NullA Can require the rows of the input matrix A to be linearly independent. (Discarding a superfluous row does not change the null space of A.)

14 Reformulating Problem 1 I input: homogeneous linear system a 1 x =0,...,a m x = 0, I output: basis b 1,...,b k for solution set Let s express this in the language of matrices: I input: matrix A = 6 I output: matrix B = a 1. a m b k b with independent rows with independent columns such that Col B =NullA Can require the rows of the input matrix A to be linearly independent. (Discarding a superfluous row does not change the null space of A.)

15 Reformulating the reformulation of Problem 1 I input: matrix A with independent rows I output: matrix B with independent columns such that Col B =NullA By Rank-Nullity Theorem, rank A +nullitya = n Because rows of A are linearly independent, rank A = m, so m +nullitya = n Requiring Col B =NullA is the same as requiring (i) Col B is a subspace of Null A (ii) dim Col B =nullitya

16 Reformulating the reformulation of Problem 1 I input: matrix A with independent rows I output: matrix B with independent columns such that Col B =NullA By Rank-Nullity Theorem, rank A +nullitya = n Because rows of A are linearly independent, rank A = m, so m +nullitya = n Requiring Col B =NullA is the same as requiring (i) Col B is a subspace of Null A =) same as requiring AB = (ii) dim Col B =nullitya

17 Reformulating the reformulation of Problem 1 I input: matrix A with independent rows I output: matrix B with independent columns such that Col B =NullA By Rank-Nullity Theorem, rank A +nullitya = n Because rows of A are linearly independent, rank A = m, so m +nullitya = n Requiring Col B =NullA is the same as requiring (i) Col B is a subspace of Null A =) same as requiring AB = (ii) dim Col B =nullitya =) same as requiring number of columns of B =nullitya same as requiring number of columns of B = n m 5

18 Reformulating the reformulation of Problem 1 I input: matrix A with independent rows I output: matrix B with independent columns such that Col B =NullA By Rank-Nullity Theorem, rank A +nullitya = n Because rows of A are linearly independent, rank A = m, so m +nullitya = n Requiring Col B =NullA is the same as requiring (i) Col B is a subspace of Null A =) same as requiring AB = (ii) dim Col B =nullitya =) same as requiring number of columns of B =nullitya same as requiring number of columns of B = n m I input: m n matrix A with independent rows h I output: matrix B with n m independent columns such that AB = 0 i 5

19 Hypothesize a procedure for reformulation of Problem 1 Problem 1: I input: m n matrix A with independent rows h I output: matrix B with n m independent columns such that AB = 0 i Define procedure null space basis(m) with this spec: I input: r n matrix M with independent rows h I output: matrix C with n r independent columns such that MC = 0 i

20 Reformulating Problem I input: independent vectors b 1,...,b k, I output: homogeneous linear system a 1 x =0,...,a m x = 0 whose solution set equals Span {b 1,...,b k } Let s express this in the language of matrices: I input: n k matrix B with independent columns I output: matrix A with independent rows such that Null A = Col B As before, Rank-Nullity Theorem implies number of rows of A +nullitya = number of columns of A As before, requiring h Null A = Col B is the same as requiring (i) AB = 0 i (ii) number of rows of A = n k I input: n k matrix B with independent rows h I output: matrix A with n k independent rows such that AB = 0 i

21 Solving Problem with the procedure for Problem 1 Problem 1: I input: m n matrix A with independent rows h I output: matrix B with n m independent columns such that AB = 0 i Define procedure null space basis(m) I input: r n matrix M with independent rows h I output: matrix C with n r independent columns such that MC = 0 i Problem : I input: n k matrix B with independent rows h I output: matrix A with n k independent rows such that AB = 0 i To solve Problem, call null space basis(b T ). h Returns matrix A T with independent columns such that B T A T = 0 i Since B T is k nh matrix, A T has n k columns. Therefore AB = 0 i and A has n k independent rows. Therefore A is solution to Problem

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013 Math Exam Sample Problems Solution Guide October, Note that the following provides a guide to the solutions on the sample problems, but in some cases the complete solution would require more work or justification

More information

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible. MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

More information

MTH 362: Advanced Engineering Mathematics

MTH 362: Advanced Engineering Mathematics MTH 362: Advanced Engineering Mathematics Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 26, 2017 1 Linear Independence and Dependence of Vectors

More information

Algorithms to Compute Bases and the Rank of a Matrix

Algorithms to Compute Bases and the Rank of a Matrix Algorithms to Compute Bases and the Rank of a Matrix Subspaces associated to a matrix Suppose that A is an m n matrix The row space of A is the subspace of R n spanned by the rows of A The column space

More information

Chapter 2 Subspaces of R n and Their Dimensions

Chapter 2 Subspaces of R n and Their Dimensions Chapter 2 Subspaces of R n and Their Dimensions Vector Space R n. R n Definition.. The vector space R n is a set of all n-tuples (called vectors) x x 2 x =., where x, x 2,, x n are real numbers, together

More information

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.

More information

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

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

More information

MAT 242 CHAPTER 4: SUBSPACES OF R n

MAT 242 CHAPTER 4: SUBSPACES OF R n MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)

More information

Math 2174: Practice Midterm 1

Math 2174: Practice Midterm 1 Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

More information

Math 308 Practice Test for Final Exam Winter 2015

Math 308 Practice Test for Final Exam Winter 2015 Math 38 Practice Test for Final Exam Winter 25 No books are allowed during the exam. But you are allowed one sheet ( x 8) of handwritten notes (back and front). You may use a calculator. For TRUE/FALSE

More information

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th. Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4

More information

Row Space and Column Space of a Matrix

Row Space and Column Space of a Matrix Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.

More information

Math 313 Chapter 5 Review

Math 313 Chapter 5 Review Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row

More information

4.9 The Rank-Nullity Theorem

4.9 The Rank-Nullity Theorem For Problems 7 10, use the ideas in this section to determine a basis for the subspace of R n spanned by the given set of vectors. 7. {(1, 1, 2), (5, 4, 1), (7, 5, 4)}. 8. {(1, 3, 3), (1, 5, 1), (2, 7,

More information

Rank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Rank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics Rank and Nullity MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives We have defined and studied the important vector spaces associated with matrices (row space,

More information

Midterm 1 Solutions Math Section 55 - Spring 2018 Instructor: Daren Cheng

Midterm 1 Solutions Math Section 55 - Spring 2018 Instructor: Daren Cheng Midterm 1 Solutions Math 20250 Section 55 - Spring 2018 Instructor: Daren Cheng #1 Do the following problems using row reduction. (a) (6 pts) Let A = 2 1 2 6 1 3 8 17 3 5 4 5 Find bases for N A and R A,

More information

Linear Algebra: Homework 7

Linear Algebra: Homework 7 Linear Algebra: Homework 7 Alvin Lin August 6 - December 6 Section 3.5 Exercise x Let S be the collection of vectors in R y that satisfy the given property. In each case, either prove that S forms a subspace

More information

Math 265 Midterm 2 Review

Math 265 Midterm 2 Review Math 65 Midterm Review March 6, 06 Things you should be able to do This list is not meant to be ehaustive, but to remind you of things I may ask you to do on the eam. These are roughly in the order they

More information

Solution: (a) S 1 = span. (b) S 2 = R n, x 1. x 1 + x 2 + x 3 + x 4 = 0. x 4 Solution: S 5 = x 2. x 3. (b) The standard basis vectors

Solution: (a) S 1 = span. (b) S 2 = R n, x 1. x 1 + x 2 + x 3 + x 4 = 0. x 4 Solution: S 5 = x 2. x 3. (b) The standard basis vectors .. Dimension In this section, we introduce the notion of dimension for a subspace. For a finite set, we can measure its size by counting its elements. We are interested in a measure of size on subspaces

More information

New concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse

New concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse Lesson 10: Rank-nullity theorem, General solution of Ax = b (A 2 R mm ) New concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse Matrix rank. matrix nullity Denition. The

More information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

Math 407: Linear Optimization

Math 407: Linear Optimization Math 407: Linear Optimization Lecture 16: The Linear Least Squares Problem II Math Dept, University of Washington February 28, 2018 Lecture 16: The Linear Least Squares Problem II (Math Dept, University

More information

The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

More information

2018 Fall 2210Q Section 013 Midterm Exam II Solution

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

More information

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer. Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]

More information

(a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii)

(a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii) . Which of the following are Vector Spaces? (i) V = { polynomials of the form q(t) = t 3 + at 2 + bt + c : a b c are real numbers} (ii) V = {at { 2 + b : a b are real numbers} } a (iii) V = : a 0 b is

More information

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 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

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

1 Last time: inverses

1 Last time: inverses MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a

More information

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

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

More information

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

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

More information

Solutions to Final Exam

Solutions to Final Exam Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns

More information

Solutions to Exam I MATH 304, section 6

Solutions to Exam I MATH 304, section 6 Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0

More information

Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence

Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the left-hand side of the coefficient matrix of a linear system A x = b which could

More information

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. 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

More information

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

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

More information

Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.

Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences. Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.. Recall that P 3 denotes the vector space of polynomials of degree less

More information

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

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

More information

Lecture 13: Row and column spaces

Lecture 13: Row and column spaces Spring 2018 UW-Madison Lecture 13: Row and column spaces 1 The column space of a matrix 1.1 Definition The column space of matrix A denoted as Col(A) is the space consisting of all linear combinations

More information

MATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS

MATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS MATH Q MIDTERM EXAM I PRACTICE PROBLEMS Date and place: Thursday, November, 8, in-class exam Section : : :5pm at MONT Section : 9: :5pm at MONT 5 Material: Sections,, 7 Lecture 9 8, Quiz, Worksheet 9 8,

More information

In Class Peer Review Assignment 2

In Class Peer Review Assignment 2 Name: Due Date: Tues. Dec. 5th In Class Peer Review Assignment 2 D.M. 1 : 7 (7pts) Short Answer 8 : 14 (32pts) T/F and Multiple Choice 15 : 30 (15pts) Total out of (54pts) Directions: Put only your answers

More information

General Vector Space (3A) Young Won Lim 11/19/12

General Vector Space (3A) Young Won Lim 11/19/12 General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version

More information

Lecture 3q Bases for Row(A), Col(A), and Null(A) (pages )

Lecture 3q Bases for Row(A), Col(A), and Null(A) (pages ) Lecture 3q Bases for Row(A), Col(A), and Null(A) (pages 57-6) Recall that the basis for a subspace S is a set of vectors that both spans S and is linearly independent. Moreover, we saw in section 2.3 that

More information

Announcements Monday, October 29

Announcements Monday, October 29 Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,

More information

Dimension. Eigenvalue and eigenvector

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,

More information

Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007

Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 You have 1 hour and 20 minutes. No notes, books, or other references. You are permitted to use Maple during this exam, but you must start with a blank

More information

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 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,

More information

Linear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008

Linear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008 Linear Algebra Chih-Wei Yi Dept. of Computer Science National Chiao Tung University November, 008 Section De nition and Examples Section De nition and Examples Section De nition and Examples De nition

More information

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions Warm-up True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

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

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/

More information

SECTION 3.3. PROBLEM 22. The null space of a matrix A is: N(A) = {X : AX = 0}. Here are the calculations of AX for X = a,b,c,d, and e. =

SECTION 3.3. PROBLEM 22. The null space of a matrix A is: N(A) = {X : AX = 0}. Here are the calculations of AX for X = a,b,c,d, and e. = SECTION 3.3. PROBLEM. The null space of a matrix A is: N(A) {X : AX }. Here are the calculations of AX for X a,b,c,d, and e. Aa [ ][ ] 3 3 [ ][ ] Ac 3 3 [ ] 3 3 [ ] 4+4 6+6 Ae [ ], Ab [ ][ ] 3 3 3 [ ]

More information

Math 308 Discussion Problems #4 Chapter 4 (after 4.3)

Math 308 Discussion Problems #4 Chapter 4 (after 4.3) Math 38 Discussion Problems #4 Chapter 4 (after 4.3) () (after 4.) Let S be a plane in R 3 passing through the origin, so that S is a two-dimensional subspace of R 3. Say that a linear transformation T

More information

Study Guide for Linear Algebra Exam 2

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

More information

CSL361 Problem set 4: Basic linear algebra

CSL361 Problem set 4: Basic linear algebra CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices

More information

MATH 2360 REVIEW PROBLEMS

MATH 2360 REVIEW PROBLEMS MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1

More information

SUMMARY OF MATH 1600

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

More information

Math 369 Exam #2 Practice Problem Solutions

Math 369 Exam #2 Practice Problem Solutions Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.

More information

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. 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

More information

Lecture 21: 5.6 Rank and Nullity

Lecture 21: 5.6 Rank and Nullity Lecture 21: 5.6 Rank and Nullity Wei-Ta Chu 2008/12/5 Rank and Nullity Definition The common dimension of the row and column space of a matrix A is called the rank ( 秩 ) of A and is denoted by rank(a);

More information

Practice Final Exam. Solutions.

Practice Final Exam. Solutions. MATH Applied Linear Algebra December 6, 8 Practice Final Exam Solutions Find the standard matrix f the linear transfmation T : R R such that T, T, T Solution: Easy to see that the transfmation T can be

More information

Basis, Dimension, Kernel, Image

Basis, Dimension, Kernel, Image Basis, Dimension, Kernel, Image Definitions: Pivot, Basis, Rank and Nullity Main Results: Dimension, Pivot Theorem Main Results: Rank-Nullity, Row Rank, Pivot Method Definitions: Kernel, Image, rowspace,

More information

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce Miderm II Solutions Problem. [8 points] (i) [4] Find the inverse of the matrix A = To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce We have A = 2 2 (ii) [2] Possibly

More information

MATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9

MATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 MATH 155, SPRING 218 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 Name Section 1 2 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 1 points. The maximum score

More information

Linear Algebra. Grinshpan

Linear Algebra. Grinshpan Linear Algebra Grinshpan Saturday class, 2/23/9 This lecture involves topics from Sections 3-34 Subspaces associated to a matrix Given an n m matrix A, we have three subspaces associated to it The column

More information

Basis, Dimension, Kernel, Image

Basis, Dimension, Kernel, Image Basis, Dimension, Kernel, Image Definitions: Pivot, Basis, Rank and Nullity Main Results: Dimension, Pivot Theorem Main Results: Rank-Nullity, Row Rank, Pivot Method Definitions: Kernel, Image, rowspace,

More information

MATH2210 Notebook 3 Spring 2018

MATH2210 Notebook 3 Spring 2018 MATH2210 Notebook 3 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 3 MATH2210 Notebook 3 3 3.1 Vector Spaces and Subspaces.................................

More information

ELE/MCE 503 Linear Algebra Facts Fall 2018

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

More information

Math 415 Exam I. Name: Student ID: Calculators, books and notes are not allowed!

Math 415 Exam I. Name: Student ID: Calculators, books and notes are not allowed! Math 415 Exam I Calculators, books and notes are not allowed! Name: Student ID: Score: Math 415 Exam I (20pts) 1. Let A be a square matrix satisfying A 2 = 2A. Find the determinant of A. Sol. From A 2

More information

Math 353, Practice Midterm 1

Math 353, Practice Midterm 1 Math 353, Practice Midterm Name: This exam consists of 8 pages including this front page Ground Rules No calculator is allowed 2 Show your work for every problem unless otherwise stated Score 2 2 3 5 4

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

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

More information

SSEA Math 51 Track Final Exam August 30, Problem Total Points Score

SSEA Math 51 Track Final Exam August 30, Problem Total Points Score Name: This is the final exam for the Math 5 track at SSEA. Answer as many problems as possible to the best of your ability; do not worry if you are not able to answer all of the problems. Partial credit

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this

More information

Chapter 2: Matrix Algebra

Chapter 2: Matrix Algebra Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry

More information

The Fundamental Theorem of Linear Algebra

The Fundamental Theorem of Linear Algebra The Fundamental Theorem of Linear Algebra Nicholas Hoell Contents 1 Prelude: Orthogonal Complements 1 2 The Fundamental Theorem of Linear Algebra 2 2.1 The Diagram........................................

More information

x 1 + 2x 2 + 3x 3 = 0 x 1 + 2x 2 + 3x 3 = 0, x 2 + x 3 = 0 x 3 3 x 3 1

x 1 + 2x 2 + 3x 3 = 0 x 1 + 2x 2 + 3x 3 = 0, x 2 + x 3 = 0 x 3 3 x 3 1 . Orthogonal Complements and Projections In this section we discuss orthogonal complements and orthogonal projections. The orthogonal complement of a subspace S is the complement that is orthogonal to

More information

We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true.

We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Dimension We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Lemma If a vector space V has a basis B containing n vectors, then any set containing more

More information

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

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

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

Lecture 22: Section 4.7

Lecture 22: Section 4.7 Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

More information

1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.

1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3. 1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is

More information

Chapter 2. General Vector Spaces. 2.1 Real Vector Spaces

Chapter 2. General Vector Spaces. 2.1 Real Vector Spaces Chapter 2 General Vector Spaces Outline : Real vector spaces Subspaces Linear independence Basis and dimension Row Space, Column Space, and Nullspace 2 Real Vector Spaces 2 Example () Let u and v be vectors

More information

x y + z = 3 2y z = 1 4x + y = 0

x y + z = 3 2y z = 1 4x + y = 0 MA 253: Practice Exam Solutions You may not use a graphing calculator, computer, textbook, notes, or refer to other people (except the instructor). Show all of your work; your work is your answer. Problem

More information

Sept. 26, 2013 Math 3312 sec 003 Fall 2013

Sept. 26, 2013 Math 3312 sec 003 Fall 2013 Sept. 26, 2013 Math 3312 sec 003 Fall 2013 Section 4.1: Vector Spaces and Subspaces Definition A vector space is a nonempty set V of objects called vectors together with two operations called vector addition

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 191 Applied Linear Algebra Lecture 16: Change of Basis Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Rank The rank of A is the dimension of the column space

More information

Solutions to Math 51 First Exam April 21, 2011

Solutions to Math 51 First Exam April 21, 2011 Solutions to Math 5 First Exam April,. ( points) (a) Give the precise definition of a (linear) subspace V of R n. (4 points) A linear subspace V of R n is a subset V R n which satisfies V. If x, y V then

More information

MATH 2030: ASSIGNMENT 4 SOLUTIONS

MATH 2030: ASSIGNMENT 4 SOLUTIONS MATH 23: ASSIGNMENT 4 SOLUTIONS More on the LU factorization Q.: pg 96, q 24. Find the P t LU factorization of the matrix 2 A = 3 2 2 A.. By interchanging row and row 4 we get a matrix that may be easily

More information

Overview. Motivation for the inner product. Question. Definition

Overview. Motivation for the inner product. Question. Definition Overview Last time we studied the evolution of a discrete linear dynamical system, and today we begin the final topic of the course (loosely speaking) Today we ll recall the definition and properties of

More information

Control Systems. Linear Algebra topics. L. Lanari

Control Systems. Linear Algebra topics. L. Lanari Control Systems Linear Algebra topics L Lanari outline basic facts about matrices eigenvalues - eigenvectors - characteristic polynomial - algebraic multiplicity eigenvalues invariance under similarity

More information

Topic 1: Matrix diagonalization

Topic 1: Matrix diagonalization Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it

More information

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

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

More information

Linear Maps and Matrices

Linear Maps and Matrices Linear Maps and Matrices Maps Suppose that V and W are sets A map F : V W is a function; that is, to every v V there is assigned a unique element w F v in W Two maps F : V W and G : V W are equal if F

More information

Linear Algebra Formulas. Ben Lee

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

More information

MATH 33A LECTURE 2 SOLUTIONS 1ST MIDTERM

MATH 33A LECTURE 2 SOLUTIONS 1ST MIDTERM MATH 33A LECTURE 2 SOLUTIONS ST MIDTERM MATH 33A LECTURE 2 SOLUTIONS ST MIDTERM 2 Problem. (True/False, pt each) Mark your answers by filling in the appropriate box next to each question. 2 3 7 (a T F

More information

Vector Spaces and Subspaces

Vector Spaces and Subspaces Vector Spaces and Subspaces Vector Space V Subspaces S of Vector Space V The Subspace Criterion Subspaces are Working Sets The Kernel Theorem Not a Subspace Theorem Independence and Dependence in Abstract

More information

MATH SOLUTIONS TO PRACTICE PROBLEMS - MIDTERM I. 1. We carry out row reduction. We begin with the row operations

MATH SOLUTIONS TO PRACTICE PROBLEMS - MIDTERM I. 1. We carry out row reduction. We begin with the row operations MATH 2 - SOLUTIONS TO PRACTICE PROBLEMS - MIDTERM I. We carry out row reduction. We begin with the row operations yielding the matrix This is already upper triangular hence The lower triangular matrix

More information

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example,

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example, Exam 2 Solutions. Let V be the set of pairs of real numbers (x, y). Define the following operations on V : (x, y) (x, y ) = (x + x, xx + yy ) r (x, y) = (rx, y) Check if V together with and satisfy properties

More information

MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix

MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix Definition: Let L : V 1 V 2 be a linear operator. The null space N (L) of L is the subspace of V 1 defined by N (L) = {x

More information