Linear Algebra 1. 3COM0164 Quantum Computing / QIP. Joseph Spring School of Computer Science. Lecture - Linear Spaces 1 1
|
|
- Darleen Cox
- 5 years ago
- Views:
Transcription
1 Linear Algebra 1 Joseph Spring School of Computer Science 3COM0164 Quantum Computing / QIP Lecture - Linear Spaces 1 1
2 Areas for Discussion Linear Space / Vector Space Definitions and Fundamental Operations Spanning Sets, Bases and Linear Independence Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 2
3 Linear Space / Vector Space Lecture - Linear Spaces 1 3
4 Fields In the following definition for a linear space we refer to a field as being composed of a set of scalars In general the fields we work with will be complex numbers and sometimes real numbers So as to maintain a sense of generality we present and define a field as follows: Lecture - Linear Spaces 1 4
5 Fields Let F denote the field 1.Addition α, β, χ F, α+ β F, the sum of α and β s.t. α + β = β + α addition is commutative (α + β) + χ = α + (β + χ) addition is associative! Scalar 0 (called zero) s.t. α F, α + 0 = α α F,! Scalar - α F s.t. α + (- α) = 0 Lecture - Linear Spaces 1 5
6 Fields 2. Multiplication α, β, χ F, αβ F, the product of α and β s.t. αβ = βα multiplication is commutative (αβ)χ = α(βχ) multiplication is associative! non zero scalar 1 F (called one) s.t. α F, α1 = α α F, α 0 ;! scalar α -1 F s.t. αα -1 = 1 3. Multiplication is distributive w.r.t. addition α, β, χ F α(β + χ) = αβ + αχ Lecture - Linear Spaces 1 6
7 Fields - Examples 1. (, +, ) 2. (, +, ) 3. (, +, ) Exercise 1. Show that 1 and 2 above are fields. We will return to 3. Lecture - Linear Spaces 1 7
8 Linear Space - Definition A Linear (Vector) Space consists of a set V of objects called vectors (denoted by v 1 >, v 2 >,, v n >) and a field F whose elements, (denoted by λ, µ, ν, ) are referred to as scalars s.t. 1.Addition Laws v 1 > + v 2 > = v 2 > + v 1 > Commutative Law ( v 1 > + v 2 >) + v 3 > = v 1 > + ( v 2 > + v 3 >) Associative Law v 1 > + 0 = 0 + v 1 > = v 1 > Additive Identity v 1 > + ( - v 1 > ) = 0 Additive Inverse Lecture - Linear Spaces 1 8
9 Linear Space - Definition 2. Multiplication α, β F, v 1 >, v 2 > V, α v 1 > V, the product of α and v 1 > s.t. α(β v 1 >) = (αβ) v 1 > 1 v 1 > = v 1 > α ( v 1 > + v 2 > ) = α v 1 > + α v 2 > multiplication by scalars is distributive w.r.t. vector addition (α + β) v 1 > = α v 1 > + β v 1 > multiplication by vectors is distributive w.r.t. scalar addition Lecture - Linear Spaces 1 9
10 Linear Space - Comments The above definition is considered a convenient characterisation of the basic objects that we wish to study The relation between a linear space V and its underlying field F is usually made clear by referring to V as a linear (vector) space over F We note that we have employed the Quantum Mechanical notation used in physics for a vector v 1 > This is sometimes referred to as a ket A linear (vector) subspace W of a linear (vector) space V is also a vector space Lecture - Linear Spaces 1 10
11 Linear Spaces - Examples 1. The set of directed line segments in the Cartesian plane forms a linear (vector) space if we define v 1 > + v 2 > to be appropriate addition and define α v 1 > to be appropriate multiplication. What were these appropriate relations? 2. The set of polynomials with real coefficients with usual algebraic addition and multiplication Lecture - Linear Spaces 1 11
12 Linear Spaces - Examples n 3. Let denote the set of all n tuples of real numbers ( n = 1, 2, 3, ) with z 1 = (x 1, x n ) and z 2 = (y 1, y n ). Let z 1 + z 2 = ( x 1 + y 1,, x n + y n ) α z = 1 ( α x 1,, α x n ) n 0 = ( 0,, 0 ) - z 1 = ( - x 1,, - x n ) n Then is a real linear space: the n dimensional real coordinate space Lecture - Linear Spaces 1 12
13 Linear Spaces - Examples 4. The most trivial linear space is the space containing one element, the zero vector Exercise 2. Show that the above are linear spaces Lecture - Linear Spaces 1 13
14 Spanning Sets Definition A spanning set for a vector space is a set of vectors { v 1 >,, v n > } each from V s.t. v > V, {α 1,, α n } each from F, s.t. v > can be written as a linear combination n v > = α i v i > i= 1 Lecture - Linear Spaces 1 14
15 Spanning Sets Example 2 A spanning set for the vector space is the set 1 v 1 > =, and v 2 > = Since α α 1 2 v > = in can be written as 2 v > = α 1 v 1 > + α 2 v 2 > Lecture - Linear Spaces 1 15
16 Spanning Sets We say that the vectors v 1 > and v 2 > span the vector space We note that a vector space may in general have many different spanning sets e.g α α v 1 > =, and v 2 > = 1 Since v > = in can be written as a linear combination of v 1 > and v 2 > α v > = v 1 > + v 2 > α α1 α Lecture - Linear Spaces 1 16
17 Linear Dependence and Independence A set of non-zero vectors { v 1 >,, v n > } is said to linearly dependent if a set of scalars {α 1,, α n } with α i 0 for at least one value of i s.t. α 1 v 1 > + α 2 v 2 > + α n v n > = 0 A set of vectors is said to be linearly independent if it is not linearly dependent Lecture - Linear Spaces 1 17
18 Bases and Dimension It can be shown that any two sets of linearly independent vectors which span a vector space V contain the same number of elements We define a basis for the vector space V to be any such linearly independent spanning set of V It can be shown that a basis always exists for a vector space V The number of elements in the basis is defined to be the dimension of the vector space V, denoted dim V We are only interested in finite dimensional vector spaces Lecture - Linear Spaces 1 18
19 Linear Dependence and Independence Question 4. Show that ( 1, -1 ), ( 1, 2 ) and ( 2,1 ) are linearly dependent Lecture - Linear Spaces 1 19
20 Linear Operators and Matrices Lecture - Linear Spaces 1 20
21 Linear Operators and Matrices Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 21
22 Linear Operators Definition A linear Operator between two linear (Vector) spaces V and W is defined to be a mapping A : V W which is linear in its inputs: A αi vi > = αia vi > i ( ) We usually write A v > for A( v > ) i Lecture - Linear Spaces 1 22
23 Identity Operator We often say that a linear operator A is defined on a vector space. By this we mean that A is a linear operator from V to V The identity operator I V is an important operator on any vector space V defined by I v > = v > V for all v >. We often drop the subscript V and write I if it is clear which vector space is under discussion Lecture - Linear Spaces 1 23
24 Zero Operator and Composition Another important operator is the zero operator which maps all vectors to the zero vector 0 v > = 0 0 Let X, Y and Z be linear (vector) spaces with A : X Y and B : Y Z be linear operators. Then the composition of B with A is denoted with BA : X Z and defined as BA( v > ) = B(A( v > )) We write BA v > as an abbreviation for BA( v > ) Lecture - Linear Spaces 1 24
25 Matrix Representations Lecture - Linear Spaces 1 25
26 Matrix Representations One of the most important topics involving linear operators acting on finite dimensional spaces is that of matrices acting on the same spaces It turns out that the linear operator and matrix viewpoints turn out to be equivalent If you are familiar with matrices then the linear operator viewpoint is an alternative way of discussing the same ideas So why not stick with matrices??? Well matrices don t extend very well to infinite dimensional spaces (/bases) which general quantum theory works With Quantum Computing we have finite dimensional bases Lecture - Linear Spaces 1 26
27 Matrix Representations Let A be an m n matrix with entries A ij Then: A n columns A 11 A A 1n A 21 A A 2n = A m1 A m 2... A mn m rows Lecture - Linear Spaces 1 27
28 Example Let A be a 2 3 matrix and B be a 3 2 matrix For example: Comments? A B = = Lecture - Linear Spaces 1 28
29 Matrix Representations We note: 1. A : n I I where I is the field of the given linear/vector space m In general I is or 2. The composition AB and BA are both defined for the given matrices. In general this is not the case. Justify! Lecture - Linear Spaces 1 29
30 Matrix Representations Let A denote a matrix acting on some linear spaace α 1 α 2 Let ψ >=. denote a vector fom the n dimensional space. α n Then A is a linear operator since: n n n A1 jµα j µ A1 jα j A1 jα j j= 1 j= 1 j= 1 α 1 µα 1 α n n n 1 α 2 µα 2 A2 jµα j µ A2 jα j A2 j α j α 2 j= 1 j= 1 j A( µ ψ >= ) A( µ. ) = A(. ) = = = µ = 1.. µ A. ( )... = = µ A ψ >... α n µα. n n n α n n Amjµα j µ Amj αj Amjα j j= 1 j= 1 j= 1 Lecture - Linear Spaces 1 30
31 Matrix Representations of Linear Operator To express a linear operator A: V W finite dimensional spaces as a matrix we proceed as follows: Let { v i >} m i=1 be a basis for V, and { w i >}n i=1 be a basis for W Since the operator A sends each v i > in V into W constants A ij field I (real or complex for us) s.t. A v j > = A ij w i > The matrix with entries A ij is said to be the matrix representation of the linear operator A Lecture - Linear Spaces 1 31
32 Matrix Representations of Linear Operator Linear Operators A Linear Space V Linear Space W v i > A ij w i > Lecture - Linear Spaces 1 32
33 Pauli Operators Lecture - Linear Spaces 1 33
34 The Pauli Operators The following 2x2 matrices frequently occur in QIP/QC and are to be found in many of the Exercises in Nielson and Chuang σ0 I, σ1 σx X i 1 0 σ2 σy Y, σ3 σz Z i Lecture - Linear Spaces 1 34
35 Summary Linear Space / Vector Space Definitions and Fundamental Operations Spanning Sets, Bases and Linear Independence Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 35
Advanced 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 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 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 informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
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 informationLS.1 Review of Linear Algebra
LS. LINEAR SYSTEMS LS.1 Review of Linear Algebra In these notes, we will investigate a way of handling a linear system of ODE s directly, instead of using elimination to reduce it to a single higher-order
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 information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
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 informationLinear Equations in Linear Algebra
Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
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 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 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 informationSystem of Linear Equations
Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given
More informationLECTURES 14/15: LINEAR INDEPENDENCE AND BASES
LECTURES 14/15: LINEAR INDEPENDENCE AND BASES MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1. Linear Independence We have seen in examples of span sets of vectors that sometimes adding additional vectors
More informationMATH 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 informationChapter 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
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
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 informationLECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS
LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems
More information6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and
6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis
More informationCSL361 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 informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationMath 1060 Linear Algebra Homework Exercises 1 1. Find the complete solutions (if any!) to each of the following systems of simultaneous equations:
Homework Exercises 1 1 Find the complete solutions (if any!) to each of the following systems of simultaneous equations: (i) x 4y + 3z = 2 3x 11y + 13z = 3 2x 9y + 2z = 7 x 2y + 6z = 2 (ii) x 4y + 3z =
More informationLinear Independence. Linear Algebra MATH Linear Algebra LI or LD Chapter 1, Section 7 1 / 1
Linear Independence Linear Algebra MATH 76 Linear Algebra LI or LD Chapter, Section 7 / Linear Combinations and Span Suppose s, s,..., s p are scalars and v, v,..., v p are vectors (all in the same space
More informationChapter 3: Theory Review: Solutions Math 308 F Spring 2015
Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)
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 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 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
More informationAFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION
4. BASES AND DIMENSION Definition Let u 1,..., u n be n vectors in V. The vectors u 1,..., u n are linearly independent if the only linear combination of them equal to the zero vector has only zero scalars;
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 informationThe scope of the midterm exam is up to and includes Section 2.1 in the textbook (homework sets 1-4). Below we highlight some of the important items.
AMS 10: Review for the Midterm Exam The scope of the midterm exam is up to and includes Section 2.1 in the textbook (homework sets 1-4). Below we highlight some of the important items. Complex numbers
More informationDuke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014
Duke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014 Linear Algebra A Brief Reminder Purpose. The purpose of this document
More informationMatrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.
Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationExam 1 - Definitions and Basic Theorems
Exam 1 - Definitions and Basic Theorems One of the difficuliies in preparing for an exam where there will be a lot of proof problems is knowing what you re allowed to cite and what you actually have to
More information1 Matrices and matrix algebra
1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns
More informationLinear Combination. v = a 1 v 1 + a 2 v a k v k
Linear Combination Definition 1 Given a set of vectors {v 1, v 2,..., v k } in a vector space V, any vector of the form v = a 1 v 1 + a 2 v 2 +... + a k v k for some scalars a 1, a 2,..., a k, is called
More informationMatrix Basic Concepts
Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
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 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 informationEndomorphisms, Automorphisms, and Change of Basis
LECTURE 15 Endomorphisms, Automorphisms, and Change of Basis We now specialize to the situation where a vector space homomorphism (a.k.a, linear transformation) maps a vector space to itself. D 15.1. Let
More information4 Vector Spaces. 4.1 Basic Definition and Examples. Lecture 10
Lecture 10 4 Vector Spaces 4.1 Basic Definition and Examples Throughout mathematics we come across many types objects which can be added and multiplied by scalars to arrive at similar types of objects.
More informationELE/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 informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationLecture 2: Linear operators
Lecture 2: Linear operators Rajat Mittal IIT Kanpur The mathematical formulation of Quantum computing requires vector spaces and linear operators So, we need to be comfortable with linear algebra to study
More informationMathematical Physics Homework 10
Georgia Institute of Technology Mathematical Physics Homework Conner Herndon November, 5 Several types of orthogonal polynomials frequently occur in various physics problems. For instance, Hermite polynomials
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationLinear Vector Spaces
CHAPTER 1 Linear Vector Spaces Definition 1.0.1. A linear vector space over a field F is a triple (V, +, ), where V is a set, + : V V V and : F V V are maps with the properties : (i) ( x, y V ), x + y
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationChapter 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 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 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 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 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 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 informationFoundations 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
More informationPhysics 342 Lecture 2. Linear Algebra I. Lecture 2. Physics 342 Quantum Mechanics I
Physics 342 Lecture 2 Linear Algebra I Lecture 2 Physics 342 Quantum Mechanics I Wednesday, January 3th, 28 From separation of variables, we move to linear algebra Roughly speaking, this is the study of
More information(II.B) Basis and dimension
(II.B) Basis and dimension How would you explain that a plane has two dimensions? Well, you can go in two independent directions, and no more. To make this idea precise, we formulate the DEFINITION 1.
More informationPhysics 342 Lecture 2. Linear Algebra I. Lecture 2. Physics 342 Quantum Mechanics I
Physics 342 Lecture 2 Linear Algebra I Lecture 2 Physics 342 Quantum Mechanics I Wednesday, January 27th, 21 From separation of variables, we move to linear algebra Roughly speaking, this is the study
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationk is a product of elementary matrices.
Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation
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 informationLinear and Bilinear Algebra (2WF04) Jan Draisma
Linear and Bilinear Algebra (2WF04) Jan Draisma CHAPTER 1 Basics We will assume familiarity with the terms field, vector space, subspace, basis, dimension, and direct sums. If you are not sure what these
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 informationTHE DIRAC EQUATION (A REVIEW) We will try to find the relativistic wave equation for a particle.
THE DIRAC EQUATION (A REVIEW) We will try to find the relativistic wave equation for a particle. First, we introduce four dimensional notation for a vector by writing x µ = (x, x 1, x 2, x 3 ) = (ct, x,
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date May 9, 29 2 Contents 1 Motivation for the course 5 2 Euclidean n dimensional Space 7 2.1 Definition of n Dimensional Euclidean Space...........
More informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationReview 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
More informationBasic 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
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 informationUnit 2, Section 3: Linear Combinations, Spanning, and Linear Independence Linear Combinations, Spanning, and Linear Independence
Linear Combinations Spanning and Linear Independence We have seen that there are two operations defined on a given vector space V :. vector addition of two vectors and. scalar multiplication of a vector
More informationENGINEERING MATH 1 Fall 2009 VECTOR SPACES
ENGINEERING MATH 1 Fall 2009 VECTOR SPACES A vector space, more specifically, a real vector space (as opposed to a complex one or some even stranger ones) is any set that is closed under an operation of
More informationMath 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 2 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 2 Fall 2010 1 / 20 Linear algebra deals with finite dimensional
More informationVector Spaces 4.5 Basis and Dimension
Vector Spaces 4.5 and Dimension Summer 2017 Vector Spaces 4.5 and Dimension Goals Discuss two related important concepts: Define of a Vectors Space V. Define Dimension dim(v ) of a Vectors Space V. Vector
More informationChapter 2. Ma 322 Fall Ma 322. Sept 23-27
Chapter 2 Ma 322 Fall 2013 Ma 322 Sept 23-27 Summary ˆ Matrices and their Operations. ˆ Special matrices: Zero, Square, Identity. ˆ Elementary Matrices, Permutation Matrices. ˆ Voodoo Principle. What is
More informationchapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS
chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader
More informationUpper triangular matrices and Billiard Arrays
Linear Algebra and its Applications 493 (2016) 508 536 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Upper triangular matrices and Billiard Arrays
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 informationLECTURE 6: LINEAR VECTOR SPACES, BASIS VECTORS AND LINEAR INDEPENDENCE. Prof. N. Harnew University of Oxford MT 2012
LECTURE 6: LINEAR VECTOR SPACES, BASIS VECTORS AND LINEAR INDEPENDENCE Prof. N. Harnew University of Oxford MT 2012 1 Outline: 6. LINEAR VECTOR SPACES, BASIS VECTORS AND LINEAR INDEPENDENCE 6.1 Linear
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 informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationMath 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 informationDefinition of Equality of Matrices. Example 1: Equality of Matrices. Consider the four matrices
IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition. Matrices 2.1 Operations with Matrices This section and the next introduce
More informationReview 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
More informationCharacterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University
Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs Sung Y. Song Iowa State University sysong@iastate.edu Notation K: one of the fields R or C X: a nonempty finite set
More informationχ 1 χ 2 and ψ 1 ψ 2 is also in the plane: αχ 1 αχ 2 which has a zero first component and hence is in the plane. is also in the plane: ψ 1
58 Chapter 5 Vector Spaces: Theory and Practice 57 s Which of the following subsets of R 3 are actually subspaces? (a The plane of vectors x =(χ,χ, T R 3 such that the first component χ = In other words,
More informationMatrix Algebra: Definitions and Basic Operations
Section 4 Matrix Algebra: Definitions and Basic Operations Definitions Analyzing economic models often involve working with large sets of linear equations. Matrix algebra provides a set of tools for dealing
More information5 Linear Transformations
Lecture 13 5 Linear Transformations 5.1 Basic Definitions and Examples We have already come across with the notion of linear transformations on euclidean spaces. We shall now see that this notion readily
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........
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 informationSpanning, linear dependence, dimension
Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R have the same size (and so does 3-space, R 3 ) That is, there is a function between
More informationLinear Algebra Notes. Lecture Notes, University of Toronto, Fall 2016
Linear Algebra Notes Lecture Notes, University of Toronto, Fall 2016 (Ctd ) 11 Isomorphisms 1 Linear maps Definition 11 An invertible linear map T : V W is called a linear isomorphism from V to W Etymology:
More informationWhat is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010
What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong sstrong@mines.edu Colorado School of Mines What is the Matrix? p. 1/20 Overview/Keywords/References Advanced
More informationOutline. Linear maps. 1 Vector addition is commutative: summands can be swapped: 2 addition is associative: grouping of summands is irrelevant:
Outline Wiskunde : Vector Spaces and Linear Maps B Jacobs Institute for Computing and Information Sciences Digital Security Version: spring 0 B Jacobs Version: spring 0 Wiskunde / 55 Points in plane The
More information