# Coordinates. Definition Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write

Save this PDF as:

Size: px
Start display at page:

Download "Coordinates. Definition Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write"

## Transcription

1 MATH10212 Linear Algebra Brief lecture notes 64 Coordinates Theorem 6.5 Let V be a vector space and let B be a basis for V. For every vector v in V, there is exactly one way to write v as a linear combination of the vectors in B. Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write v = c 1 v 1 + c 2 v c n v n Then c 1, c 2,..., c n are called the coordinates of v with respect to B, and the column vector c 1 c 2 [ v] B =. is called the coordinate vector of v with respect to B. c n Theorem 6.6 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let u and v be vectors in V and let c be a scalar. Then a. [ u + v] B = [ u] B + [ v] B b. [c u] B = c[ u] B Theorem 6.7 Let B = { v 1, v 2,..., v n } be a basis for a vector space V and let u 1,..., u k be vectors in V. Then { u 1,..., u k } is linearly independent in V if and only if {[ u 1 ] B,..., [ u k ] B } is linearly independent in R n.

2 MATH10212 Linear Algebra Brief lecture notes 65 Dimension Theorem 6.8 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. a. Any set of more than n vectors in V must be linearly dependent. b. Any set of fewer than n vectors in V cannot span V. Theorem 6.9 The Basis Theorem If a vector space V has a basis with n vectors, then every basis for V has exactly n vectors. A vector space V is called finite-dimensional if it has a basis consisting of finitely many vectors. The dimension of V, denoted by dimv, is the number of vectors in a basis for V. The dimension of the zero vector space { 0} is defined to be zero. A vector space that has no finite basis is called infinite-dimensional. Theorem 6.10 Let V be a vector space with dim V = n. Then a. Any linearly independent set in V contains at most n vectors. b. Any spanning set for V contains at least n vectors. c. Any linearly independent set of exactly n vectors in V is a basis for V. d. Any spanning set for V consisting of exactly n vectors is a basis for V. e. Any linearly independent set in V can be extended to a basis for V. f. Any spanning set for V can be reduced to a basis for V. Theorem 6.11 Let W be a subspace of a finite-dimensional vector space V. Then a. W is finite-dimensional and dimw dimv. b. dimw =dimv if and only if W = V.

3 MATH10212 Linear Algebra Brief lecture notes 66 Change of Basis Change-of-Basis Matrices Let B = { u 1,..., u n } and C = { v 1,..., v n } be bases for a vector space V. The n n matrix whose columns are the coordinate vectors [ u 1 ] C,..., [ u n ] C of the vectors in B with respect to C is denoted by P C B and is called the change-of-basis matrix from B to C. That is P C B = [[ u 1 ] C [ u 2 ] C... [ u n ] C ] Theorem 6.12 and Let be bases for a vector space V and let B = { u 1,..., u n } C = { v 1,..., v n } P C B be the change-of-basis matrix from B to C. Then a. P C B [ x] B = [ x] C for all x in V. b. P C B is the unique matrix P with the property that P [ x] B = [ x] C for all x in V. c. P C B is invertible and (P C B ) 1 = P B C. Linear Transformations A linear transformation from a vector space V to a vector space W is a mapping such that, for all u and v in V and all scalars c, 1. T ( u + v) = T ( u) + T ( v) 2. T (c u) = ct ( u) It is straightforward to show that this definition is equivalent to the requirement that T preserve all linear combinations. That is, is a linear transformation if and only if T (c 1 v c k v k ) = c 1 T ( v 1 ) + + c k T ( v k ) for all v 1,..., v k in V and scalars c 1,..., c k. Properties of Linear Transformations

4 MATH10212 Linear Algebra Brief lecture notes 67 Theorem 6.14 Let be a linear transformation. Then a. T ( 0) = 0) b. T ( v) = T ( v) for all v in V. c. T ( u v) = T ( u) T ( v) for all u and v in V. Theorem 6.15 Let be a linear transformation and let B = { v 1,..., v n } be a spanning set for V. Then T (B) = {T ( v 1 ),..., T ( v n )} spans the range of T. Composition of Linear Transformations and If T : U V S : V W are linear transformations, then the composition of S with T is the mapping S T, defined by (S T )( u) = S(T ( u)) where u is in U. Theorem If T : U V and S : V W are linear transformations, then S T : U W is a linear transformation. Inverses of Linear Transformations A linear transformation is invertible if there is a linear transformation T : W V such that T T = I V and T T = I W In this case, T is called an inverse for T.

5 MATH10212 Linear Algebra Brief lecture notes 68 Theorem 6.17 is unique. If T is an invertible linear transformation, then its inverse The Kernel and Range of a Linear Transformation Let be a linear transformation. The kernel of T, denoted ker(t ), is the set of all vectors in V that are mapped by T to 0 in W. That is, ker (T ) = { v in V : T ( v) = 0} The range of T, denoted range(t ), is the set of all vectors in W that are images of vectors in V under T. That is, range (T ) = {T ( v) : v in V } = { w in W : w = T ( v) for some v in V } Theorem Let be a linear transformation. Then a. The kernel of T is a subspace of V. b. The range of T is a subspace of W. Let be a linear transformation. The rank of T is the dimension of the range of T and is denoted by rank(t ). The nullity of T is the dimension of the kernel of T and is denoted by nullity(t ). Theorem The Rank Theorem Let be a linear transformation from a finite-dimensional vector space V into a vector space W. Then rank (T ) + nullity (T ) = dim (T ) One-to-One and Onto Linear Transformations A linear transformation is called one-to-one if T maps distinct vectors in V to distinct vectors in W. If range(t ) = W, then T is called onto.

6 MATH10212 Linear Algebra Brief lecture notes 69 Theorem A linear transformation is one-to-one if and only if ker(t ) = { 0}. Theorem Let dimv = dimw = n. Then a linear transformation is one-to-one if and only if it is onto. Theorem Let be a one-to-one linear transformation. If S = { v 1,..., v k } is a linearly independent set in V, then T (S) = {T ( v 1 ),..., T ( v k )} is a linearly independent set in W. Corollary Let dimv = dimw = n. Then a one-to-one linear transformation maps a basis for V to a basis for W. Theorem A linear transformation is invertible if and only if it is one-to-one and onto. Isomorphisms of Vector Spaces A linear transformation is called an isomorphism if it is one-to-one and onto. If V and W are two vector spaces such that there is an isomorphism from V to W, then we say that V is isomorphic to W and write V W. Theorem 6.25 Let V and W be two finite-dimensional vector spaces. Then V is isomorphic to W if and only if dimv = dimw. The Matrix of a Linear Transformation Theorem 6.26 Let V and W be two finite-dimensional vector spaces with bases B and C, respectively, where B = { v 1,..., v n }. If is a linear transformation, then the m n matrix A defined by A = [[T ( v 1 )] C [T ( v 2 )] C [T ( v n )] C ]

7 MATH10212 Linear Algebra Brief lecture notes 70 satisfies for every vector v in V. A[ v] B = [T ( v)] C Matrices of Composite and Inverse Linear Transformations Theorem Let U, V and W be finite-dimensional vector spaces with bases B, C, and D, respectively. Let T : U V and S : V W be linear transformations. Then [S T ] D B = [S] D C [T ] C B Theorem Let be a linear transformation between n- dimensional vector spaces V and W and let B and C be bases for V and W, respectively. Then T is invertible if and only if the matrix [T ] C B is invertible. In this case, ([T ] C B ) 1 = [T 1 ] B C Change of Basis and Similarity Theorem Let V be a finite-dimensional vector space with bases B and C and let T : V V be a linear transformation. Then [T ] C = P 1 [T ] B P where P is the change-of-basis matrix from C to B. Let V be a finite-dimensional vector space and let T : V V be a linear transformation. Then T is called diagonalizable if there is a basis C for V such that matrix [T ] C is a diagonal matrix. Theorem Version 4 Let A be an n n matrix and let The Fundamental Theorem of Invertible Matrices: be a linear transformation whose matrix [T ] C B with respect to bases B and C of V and W, respectively, is A. The following statements are equivalent:

8 MATH10212 Linear Algebra Brief lecture notes 71 a. A is invertible. b. A x = b has a unique solution for every b in R n. c. A x = 0 has only the trivial solution. d. The reduced row echelon form of A is I n. e. A is a product of elementary matrices. f. rank(a) = n g. nullity(a) = 0 h. The column vectors of A are linearly independent. i. The column vectors of A span R n. j. The column vectors of A form a basis for R n. k. The row vectors of A are linearly independent. l. The row vectors of A span R n. m. The row vectors of A form a basis for R n. n. det A 0 o. 0 is not an eigenvalue of A. p. T is invertible. q. T is one-to-one. r. T is onto. s. ker(t ) = { 0} t. range(t ) = W

9 MATH10212 Linear Algebra Brief lecture notes 72 Short-list of theoretical (bookwork) questions Ideally, you should know the proof of every theorem in the module: there is certainly nothing extra in it, everything belongs to the basics, bare necessities. But to make it easier for you to prepare for the exam, here is a short-list of theoretical questions, some of which will occur in the exam paper. At the exam, you do not have to reproduce the proofs in the lectures word-by-word. Common sense rules apply: if you are asked to prove something, you cannot just say...because it was proved in the lectures ; on the other hand, there is no need to prove previous lemmas and theorems on which the proof of the required bit was based in the lectures. 1. All the definitions and statements of theorems, lemmas, etc. 2. Explain why it is legitimate to use elementary row operations of the augmented matrix for solving a system of linear equations. 3. Prove that a finite system of vectors is linearly dependent if and only if (at least) one of them is a linear combination of the others. 4. Prove that e.r.o.s do not alter the span of the rows of a matrix. 5. Prove that (AB) T = B T A T. 6. Prove that A + A T and AA T are symmetric matrices. 7. Prove that if A is an invertible matrix, then a linear system A x = b has a unique solution. 8. Prove that (AB) 1 = B 1 A 1 if A 1 and B 1 exist. 9. Prove that if the r.r.e.f. of a square matrix A is I, then A is a product of elementary matrices. 10. Prove that if a square matrix A is a product of elementary matrices, then A is invertible. 11. Prove that a right or left inverse of a square matrix is a genuine two-sided inverse of it. 12. Explain why the Gauss Jordan double matrix method works for finding the inverse matrix. 13. Prove that the solution set of any homogeneous system A m n x = 0 is a subspace of R n. 14. Prove the Rank Nullity Theorem for matrices. 15. Suppose that vectors e 1,..., e k form a basis of a vector (sub)space U. Prove that then any vector in U can be uniquely represented as a linear combination of the e i. 16. Prove that every linear transformation is a matrix transformation.

10 MATH10212 Linear Algebra Brief lecture notes Prove that if a matrix is orthogonally diagonalizable, then it is symmetric. 18. Prove that the set E λ of all eigenvectors of an n n matrix A corresponding to an eigenvalue λ together with the zero vector is a subspace of R n. 19. Prove that the product of two orthogonal matrices (of the same size) is also an orthogonal matrix, and that the inverse of an orthogonal matrix is an orthogonal matrix. 20. Prove that eigenvectors for pairwise different eigenvalues are linearly independent. 21. Prove that similar matrices have the same eigenvalues, the same rank, and the same determinant. 22. Prove that det(ab) = det A det B. 23. Prove that the determinant of an orthogonal matrix is 1 or Prove that eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal. 25. Prove that the kernel of a linear transformation is a subspace of V, and the range of T is a subspace of W. 26. Prove that a linear transformation is one-to-one if and only if ker(t ) = { 0}.

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### Math 333 - Practice Exam 2 with Some Solutions

Math 333 - Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

### MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### MA106 Linear Algebra lecture notes

MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector

### Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### University of Ottawa

University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### 4.1 VECTOR SPACES AND SUBSPACES

4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following

### Linear Transformations

a Calculus III Summer 2013, Session II Tuesday, July 23, 2013 Agenda a 1. Linear transformations 2. 3. a linear transformation linear transformations a In the m n linear system Ax = 0, Motivation we can

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### Linearly Independent Sets and Linearly Dependent Sets

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### 1 Orthogonal projections and the approximation

Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### Linear Algebra Review. Vectors

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

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### Applied Linear Algebra I Review page 1

Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### 1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

### Matrices, transposes, and inverses

Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

### Row and column operations

Row and column operations It is often very useful to apply row and column operations to a matrix. Let us list what operations we re going to be using. 3 We ll illustrate these using the example matrix

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### Math 240: Linear Systems and Rank of a Matrix

Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### 1 Determinants. Definition 1

Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

### MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?

MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11