MATH 423 Linear Algebra II Lecture 12: Review for Test 1.
|
|
- Aron Taylor
- 6 years ago
- Views:
Transcription
1 MATH 423 Linear Algebra II Lecture 12: Review for Test 1.
2 Topics for Test 1 Vector spaces (F/I/S , 2.2, 2.4) Vector spaces: axioms and basic properties. Basic examples of vector spaces (coordinate vectors, matrices, polynomials, functional spaces). Subspaces. Span, spanning set. Linear independence. Basis and dimension. Various characterizations of a basis. Basis and coordinates. Change of coordinates, transition matrix. Vector space over a field.
3 Topics for Test 1 Linear transformations (F/I/S ) Linear transformations: definition and basic properties. Linear transformations: basic examples. Vector space of linear transformations. Range and null-space of a linear map. Matrix of a linear transformation. Matrix algebra and composition of linear maps. Characterization of linear maps from F n to F m. Change of coordinates for a linear operator. Isomorphism of vector spaces.
4 Sample problems for Test 1 Problem 1 (20 pts.) Let P 3 be the vector space of all polynomials (with real coefficients) of degree at most 3. Determine which of the following subsets of P 3 are subspaces. Briefly explain. (i) The set S 1 of polynomials p(x) P 3 such that p(0) = 0. (ii) The set S 2 of polynomials p(x) P 3 such that p(0) = 0 and p(1) = 0. (iii) The set S 3 of polynomials p(x) P 3 such that p(0) = 0 or p(1) = 0. (iv) The set S 4 of polynomials p(x) P 3 such that (p(0)) 2 +2(p(1)) 2 +(p(2)) 2 = 0.
5 Sample problems for Test 1 Problem 2 (20 pts.) Let V be a subspace of F(R) spanned by functions e x and ( e x. ) Let L be a 2 1 linear operator on V such that is the 3 2 matrix of L relative to the basis e x, e x. Find the matrix of L relative to the basis coshx = 1 2 (ex +e x ), sinhx = 1 2 (ex e x ). Problem 3 (25 pts.) Suppose V 1 and V 2 are subspaces of a vector space V such that dimv 1 = 5, dimv 2 = 3, dim(v 1 +V 2 ) = 6. Find dim(v 1 V 2 ). Explain your answer.
6 Sample problems for Test 1 Problem 4 (25 pts.) Consider a linear transformation T : M 2,2 (R) M 2,3 (R) given by ( ) T(A) = A for all 2 2 matrices A Find bases for the range and for the null-space of T. Bonus Problem 5 (15 pts.) Suppose V 1 and V 2 are real vector spaces, dimv 1 = m, dimv 2 = n. Let B(V 1,V 2 ) denote the subspace of F(V 1 V 2 ) consisting of bilinear functions (i.e., functions of two variables x V 1 and y V 2 that depend linearly on each variable). Prove that B(V 1,V 2 ) is isomorphic to M m,n (R).
7 Problem 1. Let P 3 be the vector space of all polynomials (with real coefficients) of degree at most 3. Determine which of the following subsets of P 3 are vector subspaces. Briefly explain. How to check whether a subset S of a vector space is a subspace? Default approach: show that S is a nonempty set closed under addition and scalar multiplication. Represent S as the span of some collection of vectors. Represent S as the null-space of a linear transformation. Represent S as the intersection of some known subspaces.
8 (i) The set S 1 of polynomials p(x) P 3 such that p(0) = 0. S 1 is not empty because it contains the zero polynomial. p 1 (0) = p 2 (0) = 0 = p 1 (0)+p 2 (0) = 0 = (p 1 +p 2 )(0) = 0. Hence S 1 is closed under addition. p(0) = 0 = rp(0) = 0 = (rp)(0) = 0. Hence S 1 is closed under scalar multiplication. Thus S 1 is a subspace of P 3.
9 (i) The set S 1 of polynomials p(x) P 3 such that p(0) = 0. Alternatively, consider a functional l : P 3 R given by l[p(x)] = p(0). It is easy to see that l is a linear functional. Clearly, S 1 is the null-space of l, hence it is a subspace of P 3.
10 (ii) The set S 2 of polynomials p(x) P 3 such that p(0) = 0 and p(1) = 0. S 2 contains the zero polynomial, S 2 is closed under addition, S 2 is closed under scalar multiplication. Thus S 2 is a subspace of P 3. Alternatively, let S 1 denote the set of polynomials p(x) P 3 such that p(1) = 0. The set S 1 is a subspace of P 3 for the same reason as S 1. Clearly, S 2 = S 1 S 1. Now the intersection of two subspaces of P 3 is also a subspace. Alternatively, S 2 is the null-space of a linear transformation L : P 3 R 2 given by L[p(x)] = (p(0),p(1)).
11 (iii) The set S 3 of polynomials p(x) P 3 such that p(0) = 0 or p(1) = 0. S 3 contains the zero polynomial, S 3 is closed under scalar multiplication, however S 3 is not closed under addition. For example, p 1 (x) = x and p 2 (x) = x 1 are in S 3 but (p 1 +p 2 )(x) = 2x 1 is not in S 3. Thus S 3 is not a subspace of P 3.
12 (iv) The set S 4 of polynomials p(x) P 3 such that (p(0)) 2 +2(p(1)) 2 +(p(2)) 2 = 0. Since coefficients of a polynomial p(x) P 3 are real, it belongs to S 4 if and only if p(0) = p(1) = p(2) = 0. Hence S 4 is the null-space of a linear transformation L : P 3 R 3 given by L[p(x)] = (p(0),p(1),p(2)). Thus S 4 is a subspace.
13 Problem 2. Let V be a subspace of F(R) spanned by functions ( e x and) e x. Let L be a linear operator on V such 2 1 that is the matrix of L relative to the basis e 3 2 x, e x. Find the matrix of L relative to the basis coshx = 1 2 (ex +e x ), sinhx = 1 2 (ex e x ). Let α denote the basis e x, e x and β denote the basis coshx, sinhx for V. Let A denote the matrix of the operator L relative to α (which is given) and B denote the matrix of L relative to β (which is to be found). By definition of the functions( coshx and ) sinhx, the transition matrix from β to α 1 1 is U = 1. It follows that B = U AU. One easily checks that 2U 2 = I. Hence U 1 = 2U so that ( )( ) ( B = U 1 AU = ) = ( )
14 Problem 3. Suppose V 1 and V 2 are subspaces of a vector space V such that dimv 1 = 5, dimv 2 = 3, dim(v 1 +V 2 ) = 6. Find dim(v 1 V 2 ). Explain your answer. We are going to show that dim(v 1 V 2 ) = dimv 1 +dimv 2 dim(v 1 +V 2 ) for any finite-dimensional subspaces V 1 and V 2. In our particular case this will imply that dim(v 1 V 2 ) = 2. First we choose a basis v 1,v 2,...,v k for the intersection V 1 V 2. The set S 0 = {v 1,v 2,...,v k } is linearly independent in both V 1 and V 2. Therefore we can extend this set to a basis for V 1 and to a basis for V 2. Let u 1,u 2,...,u m be vectors that extend S 0 to a basis for V 1 and w 1,w 2,...,w n be vectors that extend S 0 to a basis for V 2. It remains to show that v 1,...,v k,u 1,...,u m,w 1,...,w n is a basis for V 1 +V 2. Then dimv 1 = k +m, dimv 2 = k +n, dim(v 1 +V 2 ) = k +m+n, and dim(v 1 V 2 ) = k.
15 By definition, the subspace V 1 +V 2 consists of vector sums x+y, where x V 1 and y V 2. Since x is a linear combination of vectors v 1,...,v k,u 1,...,u m and y is a linear combination of vectors v 1,...,v k,w 1,...,w n, it follows that x+y is a linear combination of vectors v 1,...,v k,u 1,...,u m, w 1,...,w n. Therefore these vectors span V 1 +V 2. Now we prove that vectors v 1,...,v k,u 1,...,u m,w 1,...,w n are linearly independent. Assume r 1 v 1 + +r k v k +s 1 u 1 + +s m u m +t 1 w 1 + +t n w n = 0 for some scalars r i,s j,t l. Let x = s 1 u 1 + +s m u m, y = t 1 w 1 + +t n w n, and z = r 1 v 1 + +r k v k. Then x V 1, y V 2, and z V 1 V 2. The equality x+y+z = 0 implies that x = y z V 2 and y = x z V 1. Hence both x and y are in V 1 V 2. From this we derive that x = y = z = 0. It follows that all coefficients are zeros. Thus the vectors v 1,...,v k,u 1,...,u m,w 1,...,w n are linearly independent.
16 Problem 4. Consider a linear transformation T : M 2,2 (R) M 2,3 (R) given by ( ) T(A) = A for all 2 2 matrices A. Find bases for the range and for the null-space of T. ( ) ( ) a b a+b a a Let A =. Then T(A) = c d c +d c c ( ) = ab 1 +bb 2 +cb 3 +db 4, where B 1 =, B = ( ) ( ) ( ) , B =, B = Therefore the range of T is spanned by the matrices B 1,B 2,B 3,B 4. If ab 1 +bb 2 +cb 3 +db 4 = O for some scalars a,b,c,d R, then a+b = a = c +d = d = 0, which implies a = b = c = d = 0. Therefore B 1,B 2,B 3,B 4 are linearly independent so that they form a basis for the range of T. Also, it follows that the null-space of T is trivial.
17 Bonus Problem 5. Suppose V 1 and V 2 are real vector spaces of dimension m and n, respectively. Let B(V 1,V 2 ) denote the subspace of F(V 1 V 2 ) consisting of bilinear functions (i.e., functions of two variables x V 1 and y V 2 that depend linearly on each variable). Prove that B(V 1,V 2 ) is isomorphic to M m,n (R). Let α = [v 1,...,v m ] be an ordered basis for V 1 and β = [w 1,...,w n ] be an ordered basis for V 2. For any matrix C M m,n (R) we define a function f C : V 1 V 2 R by f C (x,y) = ([x] α ) t C[y] β for all x V 1 and y V 2. It is easy to observe that f C is bilinear. Moreover, the expression f C (x,y) depends linearly on C as well. This implies that a transformation L : M m,n (R) B(V 1,V 2 ) given by L(C) = f C is linear. The transformation L is one-to-one since the matrix C can be recovered from the function f C. Namely, if C = (c ij ), then c ij = f C (v i,w j ), 1 i m, 1 j n.
18 It remains to show that L is onto. Take any function f B(V 1,V 2 ) and vectors x V 1, y V 2. We have x = r 1 v 1 + +r m v m and y = s 1 w 1 + +s n w n for some scalars r i,s j. Using bilinearity of f, we obtain f(x,y) = f(r 1 v 1 + +r m v m,y) = = m r i f(v i,s 1 w 1 + +s n w n ) = i=1 m r i f(v i,y) i=1 m i=1 n r i s j f(v i,w j ) f(v 1,w 1 ) f(v 1,w 2 )... f(v 1,w n ) s 1 f(v = (r 1,r 2...,r m ) 2,w 1 ) f(v 2,w 2 )... f(v 2,w n ) s f(v m,w 1 ) f(v m,w 2 )... f(v m,w n ) = ([x] α ) t C [y] β j=1 so that f = f C for some matrix C M m,n (R). s n
Vector space and subspace
Vector space and subspace Math 112, week 8 Goals: Vector space, subspace. Linear combination and span. Kernel and range (null space and column space). Suggested Textbook Readings: Sections 4.1, 4.2 Week
More informationMATH 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 informationMATH 304 Linear Algebra Lecture 20: Review for Test 1.
MATH 304 Linear Algebra Lecture 20: Review for Test 1. Topics for Test 1 Part I: Elementary linear algebra (Leon 1.1 1.4, 2.1 2.2) Systems of linear equations: elementary operations, Gaussian elimination,
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationVector space and subspace
Vector space and subspace Math 112, week 8 Goals: Vector space, subspace, span. Null space, column space. Linearly independent, bases. Suggested Textbook Readings: Sections 4.1, 4.2, 4.3 Week 8: Vector
More informationMath 3013 Problem Set 6
Math 3013 Problem Set 6 Problems from 31 (pgs 189-190 of text): 11,16,18 Problems from 32 (pgs 140-141 of text): 4,8,12,23,25,26 1 (Problems 3111 and 31 16 in text) Determine whether the given set is closed
More informationAbstract Vector Spaces
CHAPTER 1 Abstract Vector Spaces 1.1 Vector Spaces Let K be a field, i.e. a number system where you can add, subtract, multiply and divide. In this course we will take K to be R, C or Q. Definition 1.1.
More informationDEF 1 Let V be a vector space and W be a nonempty subset of V. If W is a vector space w.r.t. the operations, in V, then W is called a subspace of V.
6.2 SUBSPACES DEF 1 Let V be a vector space and W be a nonempty subset of V. If W is a vector space w.r.t. the operations, in V, then W is called a subspace of V. HMHsueh 1 EX 1 (Ex. 1) Every vector space
More informationMATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field.
MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. Vector space A vector space is a set V equipped with two operations, addition V V
More informationLecture 17: Section 4.2
Lecture 17: Section 4.2 Shuanglin Shao November 4, 2013 Subspaces We will discuss subspaces of vector spaces. Subspaces Definition. A subset W is a vector space V is called a subspace of V if W is itself
More informationVector Spaces and Linear Transformations
Vector Spaces and Linear Transformations Wei Shi, Jinan University 2017.11.1 1 / 18 Definition (Field) A field F = {F, +, } is an algebraic structure formed by a set F, and closed under binary operations
More informationMATH2210 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 informationMATH 304 Linear Algebra Lecture 8: Vector spaces. Subspaces.
MATH 304 Linear Algebra Lecture 8: Vector spaces. Subspaces. Linear operations on vectors Let x = (x 1, x 2,...,x n ) and y = (y 1, y 2,...,y n ) be n-dimensional vectors, and r R be a scalar. Vector sum:
More informationVector Spaces and SubSpaces
Vector Spaces and SubSpaces Linear Algebra MATH 2076 Linear Algebra Vector Spaces & SubSpaces Chapter 4, Section 1b 1 / 10 What is a Vector Space? A vector space is a bunch of objects that we call vectors
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) March 5, 2015
Midterm 1 Advanced Linear Algebra Math 4377 / 638 (Spring 215) March 5, 215 2 points 1. Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain
More informationMidterm solutions. (50 points) 2 (10 points) 3 (10 points) 4 (10 points) 5 (10 points)
Midterm solutions Advanced Linear Algebra (Math 340) Instructor: Jarod Alper April 26, 2017 Name: } {{ } Read all of the following information before starting the exam: You may not consult any outside
More informationMATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited).
MATH 311-504 Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). Definition. A mapping f : V 1 V 2 is one-to-one if it maps different elements from V 1 to different
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationThe 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 informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 2.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined
More informationAbstract Vector Spaces and Concrete Examples
LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.
More information1 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 informationMATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
More informationMath 4377/6308 Advanced Linear Algebra
1.3 Subspaces Math 4377/6308 Advanced Linear Algebra 1.3 Subspaces Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377 Jiwen He, University of Houston
More informationLinear Algebra (Math-324) Lecture Notes
Linear Algebra (Math-324) Lecture Notes Dr. Ali Koam and Dr. Azeem Haider September 24, 2017 c 2017,, Jazan All Rights Reserved 1 Contents 1 Real Vector Spaces 6 2 Subspaces 11 3 Linear Combination and
More informationECS130 Scientific Computing. Lecture 1: Introduction. Monday, January 7, 10:00 10:50 am
ECS130 Scientific Computing Lecture 1: Introduction Monday, January 7, 10:00 10:50 am About Course: ECS130 Scientific Computing Professor: Zhaojun Bai Webpage: http://web.cs.ucdavis.edu/~bai/ecs130/ Today
More informationDr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science
Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated
More informationSecond Exam. Math , Spring March 2015
Second Exam Math 34-54, Spring 25 3 March 25. This exam has 8 questions and 2 pages. Make sure you have all pages before you begin. The eighth question is bonus (and worth less than the others). 2. This
More informationReview 1 Math 321: Linear Algebra Spring 2010
Department of Mathematics and Statistics University of New Mexico Review 1 Math 321: Linear Algebra Spring 2010 This is a review for Midterm 1 that will be on Thursday March 11th, 2010. The main topics
More informationLinear Algebra Lecture Notes-I
Linear Algebra Lecture Notes-I Vikas Bist Department of Mathematics Panjab University, Chandigarh-6004 email: bistvikas@gmail.com Last revised on February 9, 208 This text is based on the lectures delivered
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationRow Space, Column Space, and Nullspace
Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space
More informationLinear 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 informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN-3 = 978--68-78376-2,
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationMATH 423 Linear Algebra II Lecture 11: Change of coordinates (continued). Isomorphism of vector spaces.
MATH 423 Linear Algebra II Lecture 11: Change of coordinates (continued). Isomorphism of vector spaces. Change of coordinates Let V be a vector space of dimension n. Let v 1,v 2,...,v n be a basis for
More informationMath 113 Practice Final Solutions
Math 113 Practice Final Solutions 1 There are 9 problems; attempt all of them. Problem 9 (i) is a regular problem, but 9(ii)-(iii) are bonus problems, and they are not part of your regular score. So do
More informationMath 110, Spring 2015: Midterm Solutions
Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make
More informationMAT 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 informationWorksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases
Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation
More informationGENERAL VECTOR SPACES AND SUBSPACES [4.1]
GENERAL VECTOR SPACES AND SUBSPACES [4.1] General vector spaces So far we have seen special spaces of vectors of n dimensions denoted by R n. It is possible to define more general vector spaces A vector
More informationMath 544, Exam 2 Information.
Math 544, Exam 2 Information. 10/12/10, LC 115, 2:00-3:15. Exam 2 will be based on: Sections 1.7, 1.9, 3.2, 3.3, 3.4; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/544fa10/544.html)
More informationStudy 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 informationMATH 110: LINEAR ALGEBRA HOMEWORK #2
MATH 11: LINEAR ALGEBRA HOMEWORK #2 1.5: Linear Dependence and Linear Independence Problem 1. (a) False. The set {(1, ), (, 1), (, 1)} is linearly dependent but (1, ) is not a linear combination of the
More informationVector Spaces - Definition
Vector Spaces - Definition Definition Let V be a set of vectors equipped with two operations: vector addition and scalar multiplication. Then V is called a vector space if for all vectors u,v V, the following
More informationHonours Algebra 2, Assignment 8
Honours Algebra, Assignment 8 Jamie Klassen and Michael Snarski April 10, 01 Question 1. Let V be the vector space over the reals consisting of polynomials of degree at most n 1, and let x 1,...,x n be
More informationObjective: Introduction of vector spaces, subspaces, and bases. Linear Algebra: Section
Objective: Introduction of vector spaces, subspaces, and bases. Vector space Vector space Examples: R n, subsets of R n, the set of polynomials (up to degree n), the set of (continuous, differentiable)
More informationVector Spaces. (1) Every vector space V has a zero vector 0 V
Vector Spaces 1. Vector Spaces A (real) vector space V is a set which has two operations: 1. An association of x, y V to an element x+y V. This operation is called vector addition. 2. The association of
More informationWorksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases
Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation
More informationChapter 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 informationLecture 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 informationMATH 260 LINEAR ALGEBRA EXAM II Fall 2013 Instructions: The use of built-in functions of your calculator, such as det( ) or RREF, is prohibited.
MAH 60 LINEAR ALGEBRA EXAM II Fall 0 Instructions: he use of built-in functions of your calculator, such as det( ) or RREF, is prohibited ) For the matrix find: a) M and C b) M 4 and C 4 ) Evaluate the
More informationDefinition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition
6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition
More informationChapter 3. Vector spaces
Chapter 3. Vector spaces Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/22 Linear combinations Suppose that v 1,v 2,...,v n and v are vectors in R m. Definition 3.1 Linear combination We say
More informationwhich are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar.
It follows that S is linearly dependent since the equation is satisfied by which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. is
More information4 Chapter 4 Lecture Notes. Vector Spaces and Subspaces
Math 2040 Matrix Theory and Linear Algebra II 4 hapter 4 Lecture Notes. Vector Spaces and Subspaces 4.1 Vector Spaces and Subspaces 1. Notation: The symbol means the empty set. The symbol means is an element
More information17. C M 2 (C), the set of all 2 2 matrices with complex entries. 19. Is C 3 a real vector space? Explain.
250 CHAPTER 4 Vector Spaces 14. On R 2, define the operation of addition by (x 1,y 1 ) + (x 2,y 2 ) = (x 1 x 2,y 1 y 2 ). Do axioms A5 and A6 in the definition of a vector space hold? Justify your answer.
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationNOTES (1) FOR MATH 375, FALL 2012
NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More information3.2 Subspace. Definition: If S is a non-empty subset of a vector space V, and S satisfies the following conditions: (i).
. ubspace Given a vector spacev, it is possible to form another vector space by taking a subset of V and using the same operations (addition and multiplication) of V. For a set to be a vector space, it
More informationLecture 16: 9.2 Geometry of Linear Operators
Lecture 16: 9.2 Geometry of Linear Operators Wei-Ta Chu 2008/11/19 Theorem 9.2.1 If T: R 2 R 2 is multiplication by an invertible matrix A, then the geometric effect of T is the same as an appropriate
More informationMATH 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
More information2: LINEAR TRANSFORMATIONS AND MATRICES
2: LINEAR TRANSFORMATIONS AND MATRICES STEVEN HEILMAN Contents 1. Review 1 2. Linear Transformations 1 3. Null spaces, range, coordinate bases 2 4. Linear Transformations and Bases 4 5. Matrix Representation,
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 02 Vector Spaces, Subspaces, linearly Dependent/Independent of
More information0.2 Vector spaces. J.A.Beachy 1
J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a
More informationMath 321: Linear Algebra
Math 32: Linear Algebra T. Kapitula Department of Mathematics and Statistics University of New Mexico September 8, 24 Textbook: Linear Algebra,by J. Hefferon E-mail: kapitula@math.unm.edu Prof. Kapitula,
More informationMATH 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 information2018 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 informationSYMBOL EXPLANATION EXAMPLE
MATH 4310 PRELIM I REVIEW Notation These are the symbols we have used in class, leading up to Prelim I, and which I will use on the exam SYMBOL EXPLANATION EXAMPLE {a, b, c, } The is the way to write the
More information6. The scalar multiple of u by c, denoted by c u is (also) in V. (closure under scalar multiplication)
Definition: A subspace of a vector space V is a subset H of V which is itself a vector space with respect to the addition and scalar multiplication in V. As soon as one verifies a), b), c) below for H,
More informationMATH 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 informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationGeneral 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 informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationMATH SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER Given vector spaces V and W, V W is the vector space given by
MATH 110 - SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER 2009 GSI: SANTIAGO CAÑEZ 1. Given vector spaces V and W, V W is the vector space given by V W = {(v, w) v V and w W }, with addition and scalar
More information1 Linear transformations; the basics
Linear Algebra Fall 2013 Linear Transformations 1 Linear transformations; the basics Definition 1 Let V, W be vector spaces over the same field F. A linear transformation (also known as linear map, or
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationLecture 9: Vector Algebra
Lecture 9: Vector Algebra Linear combination of vectors Geometric interpretation Interpreting as Matrix-Vector Multiplication Span of a set of vectors Vector Spaces and Subspaces Linearly Independent/Dependent
More informationLecture 11: Vector space and subspace
Lecture : Vector space and subspace Vector space. R n space Definition.. The space R n consists of all column vector v with n real components, i.e. R n = { v : v = [v,v 2,...,v n ] T, v j R,j =,2,...,n
More informationNAME MATH 304 Examination 2 Page 1
NAME MATH 4 Examination 2 Page. [8 points (a) Find the following determinant. However, use only properties of determinants, without calculating directly (that is without expanding along a column or row
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationMATH Linear Algebra
MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
More information1 Last time: multiplying vectors matrices
MATH Linear algebra (Fall 7) Lecture Last time: multiplying vectors matrices Given a matrix A = a a a n a a a n and a vector v = a m a m a mn Av = v a a + v a a v v + + Rn we define a n a n a m a m a mn
More informationMath 205, Summer I, Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b. Chapter 4, [Sections 7], 8 and 9
Math 205, Summer I, 2016 Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b Chapter 4, [Sections 7], 8 and 9 4.5 Linear Dependence, Linear Independence 4.6 Bases and Dimension 4.7 Change of Basis,
More informationModern Control Systems
Modern Control Systems Matthew M. Peet Arizona State University Lecture 2: Mathematical Preliminaries Mathematics - Sets and Subsets Definition 1. A set, D, is any collection of elements, d. Denoted d
More information2. 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 informationWe 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= c. = c. c 2. We can find apply our general formula to find the inverse of the 2 2 matrix A: A 1 5 4
. In each part, a basis B of R is given (you don t need to show B is a basis). Find he B-coordinate of the vector v. (a) B {, }, v Solution.(5 points) We have: + Therefore, the B-coordinate of v is equal
More informationMath 3108: Linear Algebra
Math 3108: Linear Algebra Instructor: Jason Murphy Department of Mathematics and Statistics Missouri University of Science and Technology 1 / 323 Contents. Chapter 1. Slides 3 70 Chapter 2. Slides 71 118
More informationSept. 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 informationYORK 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 informationLineaire algebra 1 najaar Oefenopgaven. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra
Lineaire algebra 1 najaar 2008 ontleend aan: Oefenopgaven Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra Charles W. Curtis, Linear Algebra: An Introductory Approach Exercise
More informationOn-Line Geometric Modeling Notes VECTOR SPACES
On-Line Geometric Modeling Notes VECTOR SPACES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis These notes give the definition of
More information2.2. Show that U 0 is a vector space. For each α 0 in F, show by example that U α does not satisfy closure.
Hints for Exercises 1.3. This diagram says that f α = β g. I will prove f injective g injective. You should show g injective f injective. Assume f is injective. Now suppose g(x) = g(y) for some x, y A.
More information