Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
|
|
- Valentine Sanders
- 5 years ago
- Views:
Transcription
1 Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a a 1j... a 1n a 21. a a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column of a matrix A is refered to as (A) ij. EXAMPLE: Algebra 2017/ A zero matrix is a matrix, written 0, whose entries are all zero. A square matrix has the same number of rows than columns. In general (m n), matrices are rectangular. The (main) diagonal of a matrix, or its diagonal entries, are the entries A diagonal matrix has all its nondiagonal entries equal to zero Algebra 2017/
2 A matrix is upper triangular if all its elements under the diagonal are zero A matrix is lower triangular if all its elements over the diagonal are zero The set of all possible matrices of dimension (m n) whose entries are real numbers is refered to as R m n The set of all possible matrices of dimension (m n) whose entries are complex numbers is refered to as C m n K 3 2 Algebra 2017/ OPERATIONS: Only for matrices with the same dimensions: Equality. Two matrices are equal if and only if their corresponding entries are equal Addition. A matrix whose entries are the sum of the corresponding entries of the matrices Algebra 2017/
3 Scalar Multiplication. A matrix whose entries are the corresponding entries of the matrix multiplied by the scalar PROPERTIES: Let A, B and C be matrices of K m n and λ, µ K: A + B B + A A+(B+C)(A+B)+C A + 0 A λ (A+B) λ A+λ B (λ+µ) A λ A+µ A λ (µ A) (λ µ) A Algebra 2017/ Matrix Multiplication K p K n K m One wonders: Does C exist C x A B x x K p? PROBLEM: What dimensions would C have? Algebra 2017/
4 If we write B [ b 1 b 2... b p and x x 1. x p, then: Bx x 1 b 1 + x 2 b x p b p A(Bx) Algebra 2017/ Let A be an (m n) matrix and let B be an (n p) matrix with columns b 1, b 2,..., b p. The matrix product of A by B is the (m p) matrix AB whose columns are Ab 1, Ab 2,..., Ab p. That is, AB A [ b 1 b 2... b p [ Ab 1 Ab 2... Ab p Warning: The dimensions of the matrices involved in a product must verify A B C ( ) ( ) ( ) Algebra 2017/
5 EXAMPLE: [ [ ( ) ( ) ( ) [ [ [ [ [ Algebra 2017/ Row-Column Rule for computing AB Consider A K m n, and B [ b 1... b p K n p such that (A) ik a ik, and (B) kj b kj. That is, AB [ Ab 1 Ab j Ab p {}}{ a 11 a a 1n 1.. b 1j a i1 a i2... a in b 2j.... i b. (AB) ij nj a m1 a m2... a mn (AB) ij a i1 a i2... a in m b 1j b 2j. b nj k a ik b kj Algebra 2017/
6 EXAMPLE: [ [ [ 1st row 3rd column ( 1, 3 ) entry [ [ [ 2nd row 1st column ( 2, 1 ) entry Algebra 2017/ PROBLEM: Find the 2nd row of AB. AB PROBLEM: Compute [ [ Algebra 2017/
7 PROPERTIES: Let A be an (m n) matrix, and B and C matrices of appropriate dimensions: A(BC) (AB)C A(B + C) AB + AC (B + C)A BA + CA µ (AB) (µ A) B A (µ B) µ K I m A A A I n where I k is the (k k) identity matrix 4.3 Algebra 2017/ WARNING: In general, AB BA EXPANSION AXIS X ROTATION 30 1st EXPANSION + 2nd ROTATION [ 2 0 B A AB 3 ROTATION 30 o EXPANSION AXIS X 1st ROTATION + 2nd EXPANSION [ A B BA Algebra 2017/
8 WARNING: In general, AB AC / B C ROTATION π/2 PROJECTION in X 1st ROTATION + 2nd PROJECTION B A AB REFLECTION x+y 0 PROJECTION in X 1st REFLECTION + 2nd PROJECTION C A AC Algebra 2017/ WARNING: In general, AB 0 / A 0 or B 0 PROJECTION in X PROJECTION in Y 1st X-PROJECTION + 2nd Y-PROJECTION B A AB WARNING: In general, A 2 0 / A 0 [ A [, A A 2 If two square matrices verify that AB BA, we say that A and B commute with each other. Algebra 2017/
9 The kth power of a matrix is defined: A k A A A A }{{} k times This only makes sense if A is a nonnegative integer. matrix and k is a For convenience, we define A 0 I. PROBLEM: Compute 4.4 Algebra 2017/ Transpose of a Matrix The transpose of an (m n) matrix A is the (n m) matrix A T whose columns are the rows of A. That is, EXAMPLE: B (A T ) ij (A) ji B T EXAMPLE: A symmetric matrix verifies A T A. An antisymmetric matrix verifies A T A. PROBLEM: Provide examples of (anti)symmetric matrices. Algebra 2017/
10 PROPERTIES: Let A and B be matrices of appropriate dimensions and µ K: (A T ) T A (µ A) T µ (A T ) (A + B) T A T + B T (AB) T B T A T Proof: Let be A K m n and B K n q ( (AB) T ) ij PROBLEM: Prove that (ABC) T C T B T A T. 4.7 Algebra 2017/ Conjugate Transpose of a Matrix The conjugate transpose of an (m n) matrix A is the (n m) matrix A, or A H, whose elements verify: (A ) ij (A) ji. EXAMPLE: 5 2 i B i B A [ a 1 a 2 a n A Algebra 2017/
11 PROPERTIES: Let A and B be matrices of appropriate dimensions and µ K: (A ) A (A + B) A + B (µ A) µ (A ) (AB) B A A A T if and only if A is a real matrix. A Hermitian matrix verifies A A. An antihermitian matrix verifies A A. PROBLEM: Provide examples of (anti)hermitian matrices. 4.8 Algebra 2017/ Inverse of a Matrix A square (n n) matrix A is invertible, or nonsingular, if there exists a matrix B such that AB I n A noninvertible or singular matrix has no inverse. 2 5 EXAMPLE: This matrix is invertible: A Because C verifies AC Algebra 2017/
12 [ cos φ sin φ EXAMPLE: This matrix is invertible: A sin φ cos φ A A 1 Thus, A 1 EXAMPLE: Matrix B has no inverse and is, therefore, a singular matrix: B 4.9 Algebra 2017/ Theorem 4.1. If A is an invertible (n n) matrix, then the equation Axb has the unique solution xa 1 b, b K n. Proof: That x A 1 b is a solution b can be checked by a mere substitution: As it has a solution b A must have a pivot in every row. A square No free variables Warning: Algebra 2017/
13 Theorem 4.2. Let A and B be (n n) matrices. Then: AB I BA I Proof: ( AB I BA I ) Suppose that BA X Let s define M I X [ m 1 m 2 m n. As That is, But now, Leading to Algebra 2017/ Theorem 4.3. If A is an invertible matrix, then A 1 is invertible and (A 1 ) 1 A. Proof: Theorem 4.4. If exists, the inverse of a matrix is unique. Proof: Let A be an invertible matrix, and B a matrix such that AB I (that is, B A 1 ). Suppose there exists C such that AC I (in other words, suppose that A has another inverse). Algebra 2017/
14 Theorem 4.5. If A is invertible, A T is also invertible and (A T ) 1 (A 1 ) T. Theorem 4.6. If A is invertible, A is also invertible and (A ) 1 (A 1 ). Proof: EXAMPLE: [ 1+i 1+2i 1 1 i [ 1 i 1 2i i then, [ 1 + i i 1 i 1 Algebra 2017/ Theorem 4.7. If A and B are invertible (n n) matrices, then AB is invertible and (AB) 1 B 1 A 1. Proof: EXAMPLE: Consider the linear transformations: A ROTATE B EXPAND. Then, AB (in this order!) and the inverse is (AB) PROBLEM: If A, B and C are nonsingular matrices of equal size, show that (ABC) 1 C 1 B 1 A 1. Algebra 2017/
15 An elementary matrix is one that is obtained by performing one elementary row operation on an identity matrix. EXAMPLE: E E E Notice: These matrices have a clear geometrical interpretation. They correspond to Algebra 2017/ Theorem 4.8. If an elementary row operation if performed on an (m n) matrix A, the resulting matrix can be written as EA, where E is the (m m) elementary matrix created by performing the same operation on I m. EXAMPLE: Consider the (3 2) matrix A a d b e c f I E 1 ( r 3 5 r 3 ) E 1 A a d b e c f Algebra 2017/
16 I E 2 ( r 2 r 3 ) E 2 A a d b e c f I E 3 ( r 2 r 2 4r 1 ) E 3 A a d b e c f A A A Algebra 2017/ Theorem 4.9. Every elementary matrix E is invertible and its inverse E 1 is the elementary matrix corresponding to the row operation that transforms E back into I. EXAMPLE: The matrix E 1 multiplies the 3rd row by five: E Its inverse E1 1 is the matrix that divides the 3rd row by five: E 1 1 Check: E 1 E 1 1 I 4.12 PROBLEM: Find the matrices E 1 2 and E 1 3. Algebra 2017/
17 Theorem An (n n) matrix A is invertible if and only if A is row equivalent to I n. In this case, any sequence of elementary row operations that transforms A into I n also transforms I n in A 1. Proof: A invertible Then, A 1 E p E p 1... E 2 E 1 and, in fact, 4.14 Algebra 2017/ An Algorithm for finding A 1 Construct the matrix [ A I Find its reduced echelon form. If this matrix has the form [ I B, then A 1 B. Otherwise, A does not have an inverse. EXAMPLE: }{{} }{{} }{{}}{{} A I Algebra 2017/
18 }{{}}{{} I }{{}}{{} }{{}}{{} A 1 Algebra 2017/ PROBLEM: If exists, find the inverse of the matrix C [ C I 4.16 Check: C C 1 Algebra 2017/
19 Theorem (The Square Matrix Theorem) If A K n n, the following statements are equivalent: 1. A is an invertible matrix. 2. There exists C K n n such that AC I n. 3. There exists D K n n such that DA I n. 4. A is row equivalent to I n. 5. A has n pivots. 6. The equation Ax 0 has only the trivial solution. 7. The columns/rows of A are linearly independent. 8. The equation Ax b has a (unique) solution b K n. 9. The columns/rows of A span K n. 10. The columns/rows of A form a basis of K n 11. A T is invertible. 12. A is invertible. 13. The linear transformation x Ax is bijective. 14. Col A Row A K n 15. dim Col A dim Row A n 16. rank A n 17. Nul A {0} 18. dim Nul A 0 A transformation T : K n K n is called invertible if there exists a transformation S : K n K n such that } S(T (x)) x x K n. T (S(x)) x The transformation S is called the inverse of T. Theorem Let T : K n K n be a linear transformation and A its canonical matrix. T is invertible if and only if A is nonsingular. In this case, S(x) A 1 x Algebra 2017/
20 4.3. Partitioned (or Block) Matrices EXAMPLE: A A where A 11, A 12, A 13 A 21, A 22, A 23 Algebra 2017/ EXAMPLE: Social web of 6 persons in 3 groups Adjacency Matrix M M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 Algebra 2017/
21 EXAMPLE: Jefferson High School Algebra 2017/ EXAMPLE: Trade share matrix between countries Algebra 2017/
22 PROPERTIES: Addition: Matrices of equal size and identical partition can be summed block by block: A + B [ A11 A 12 A 13 A 21 A 22 A 23 [ B11 B + 12 B 13 B 21 B 22 B 23 Scalar Multiplication: [ A11 A λa λ 12 A 13 A 21 A 22 A 23 Algebra 2017/ Transpose of a matrix: [ A11 A A 12 A 13 A 21 A 22 A 23 A T A T 11 A T 21 A T 12 A T 22 A T 13 A T 23 Conjugate transpose of a matrix: A 11 A 21 A11 A A 12 A 13 A A A 21 A 22 A A 22 A 13 A 23 EXAMPLE: A A T Algebra 2017/
23 Multiplication of partitioned matrices: Two matrices A and B of respective dimensions (m n) and (n p) are conformable for block multiplication when the number of columns of each partition of A is equal to the number of rows of the corresponding partition of B. AB [ A11 A 12 A 21 A [ B11 B (Attention: ) Algebra 2017/ Concentrate on the dimensions of the blocks: (2 3) ( ) ( ) (3 5) (5 2) ( ) ( ) ( ) (2 3)(3 2) + ( )( ) ( )( ) + ( )( ) (2 2) + ( ) ( ) + ( ) ( ) ( ) [ ( ) Algebra 2017/
24 EXAMPLE: Let A be a block upper triangular matrix: A A11 A A 22 Assuming that A is invertible, A 11 is (p p) and A 22 is (q q), find a formula for A 1. Call B A 1. Partition B in such a way that we can write: A11 A AB 12 B11 B 12 I 0. 0 A 22 B 21 B 22 0 I The dimensions of the matrices involved are: (p p) ( ) ( ) ( ) ( ) (q q) ( ) ( ) [ ( ) ( ) ( ) ( ). Algebra 2017/ The equation can be written: [ I 0 0 I. Equating components, we obtain: (a) I (b) 0 (c) 0 (d) I We have to solve 4 matrix equations, which represent a linear system of (p + q) 2 equations with (p + q) 2 unknowns. Algebra 2017/
25 (d) (c) (a) (b) Obtaining, A 1. Algebra 2017/ Theorem A block diagonal matrix is invertible if and only if each of the diagonal blocks is invertible. Proof: The case of two blocks follows from the above result when A C C C nn Algebra 2017/
26 Theorem A diagonal matrix is invertible if and only if none of its diagonal elements is zero. 1 a a a nn 4.19 PROBLEM: Determine under what conditions the following matrix is invertible and, in that case, find its inverse: Im 0. A I n Algebra 2017/ Determinants Given an (m n) matrix A, we define the minor A ij as the ((m 1) (n 1)) matrix obtained by removing the ith row and the jth column of the matrix A. EXAMPLE: A Let A be an (n n) matrix whose entry (A) ij a ij. We define the determinant of A as det A A n ( 1) j+1 a 1j det A 1j j1 n a 1j C 1j, j1 where C ij ( 1) i+j det A ij is refered to as the ij cofactor of A. Algebra 2017/
27 Theorem The determinant of a square matrix A can be expressed as the cofactor expansion along any row of the matrix n n ( ) det A ( 1) k+j along the a kj det A kj a kj C kj j1 j1 kth row WARNING: Algebra 2017/ EXAMPLE: 1st row: det nd row: Algebra 2017/
28 Theorem If A is an (n n) triangular matrix, its determinant is the product of its diagonal entries. det a a a a 44 0 a 55 Algebra 2017/ Theorem Let A be an (n n) matrix. If we obtain a matrix B, By adding to a row of A the multiple of another row, det B det A. By multiplying one row of A by λ, det B λ det A. By interchanging two rows of A, det B det A. EXAMPLE: Algebra 2017/
29 Theorem Let A be a square matrix and U an echelon matrix obtained from A by adding multiples of rows and r row interchanges (but without multiplying any row by a scalar!). Then, 0 if A is not invertible det A ) ( 1) r if A is invertible ( product of the pivots 4.20 Proof: Note: This would add a new statement to theorem 4.11: 19. The determinant of A is nonzero. Algebra 2017/ WARNING: In general, Check theorem 4.17! A B / det A det B. WARNING: In general, det(a + B) det A + det B. EXAMPLE: If it was true, all determinants would be zero: ([ a b a 0 det det c d b ) c 0 0 d Algebra 2017/
30 Theorem If A and B are square matrices, det(ab) det A det B. Theorem If A is a square matrix, A T A and A A Proof: For elementary matrices, it s easy to see that E E T. If we obtain an echelon form of a matrix A: Leading to Now, as U is a triangular matrix, U T U and, consequently 4.23 Algebra 2017/
MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
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
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
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 informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationMatrix Algebra. Matrix Algebra. Chapter 8 - S&B
Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number
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 informationUndergraduate Mathematical Economics Lecture 1
Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses
More informationElementary Row Operations on Matrices
King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix
More informationICS 6N Computational Linear Algebra Matrix Algebra
ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
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 informationGraduate Mathematical Economics Lecture 1
Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationLinear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay
Linear Algebra Summary Based on Linear Algebra and its applications by David C. Lay Preface The goal of this summary is to offer a complete overview of all theorems and definitions introduced in the chapters
More informationChapter 2: Matrix Algebra
Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry
More information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More informationMatrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc.
2 Matrix Algebra 2.1 MATRIX OPERATIONS MATRIX OPERATIONS m n If A is an matrixthat is, a matrix with m rows and n columnsthen the scalar entry in the ith row and jth column of A is denoted by a ij and
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationMAT 2037 LINEAR ALGEBRA I web:
MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationI = i 0,
Special Types of Matrices Certain matrices, such as the identity matrix 0 0 0 0 0 0 I = 0 0 0, 0 0 0 have a special shape, which endows the matrix with helpful properties The identity matrix is an example
More informationLinear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More informationChapter 4. Determinants
4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationSection 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.
Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationMatrix Arithmetic. j=1
An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMatrices. In this chapter: matrices, determinants. inverse matrix
Matrices In this chapter: matrices, determinants inverse matrix 1 1.1 Matrices A matrix is a retangular array of numbers. Rows: horizontal lines. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationMTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~
MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is One-to-one Onto Question
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More information1 Last time: determinants
1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)
More informationTOPIC III LINEAR ALGEBRA
[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
More informationChapter 1 Matrices and Systems of Equations
Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationThings we can already do with matrices. Unit II - Matrix arithmetic. Defining the matrix product. Things that fail in matrix arithmetic
Unit II - Matrix arithmetic matrix multiplication matrix inverses elementary matrices finding the inverse of a matrix determinants Unit II - Matrix arithmetic 1 Things we can already do with matrices equality
More informationJim Lambers MAT 610 Summer Session Lecture 1 Notes
Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationMATH10212 Linear Algebra Lecture Notes
MATH10212 Linear Algebra Lecture Notes Last change: 23 Apr 2018 Textbook Students are strongly advised to acquire a copy of the Textbook: D. C. Lay. Linear Algebra and its Applications. Pearson, 2006.
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationCHAPTER 6. Direct Methods for Solving Linear Systems
CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) 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 and b are real numbers. 1
More informationECON 186 Class Notes: Linear Algebra
ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).
More informationHomework Set #8 Solutions
Exercises.2 (p. 19) Homework Set #8 Solutions Assignment: Do #6, 8, 12, 14, 2, 24, 26, 29, 0, 2, 4, 5, 6, 9, 40, 42 6. Reducing the matrix to echelon form: 1 5 2 1 R2 R2 R1 1 5 0 18 12 2 1 R R 2R1 1 5
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University February 6, 2018 Linear Algebra (MTH
More informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More informationMATH10212 Linear Algebra Lecture Notes
MATH1212 Linear Algebra Lecture Notes Textbook Students are strongly advised to acquire a copy of the Textbook: D. C. Lay. Linear Algebra and its Applications. Pearson, 26. ISBN -521-28713-4. Other editions
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 informationENGR-1100 Introduction to Engineering Analysis. Lecture 21. Lecture outline
ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationENGR-1100 Introduction to Engineering Analysis. Lecture 21
ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility
More informationMatrices and Determinants
Chapter1 Matrices and Determinants 11 INTRODUCTION Matrix means an arrangement or array Matrices (plural of matrix) were introduced by Cayley in 1860 A matrix A is rectangular array of m n numbers (or
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More informationDigital Workbook for GRA 6035 Mathematics
Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationApplied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University
Applied Matrix Algebra Lecture Notes Section 22 Gerald Höhn Department of Mathematics, Kansas State University September, 216 Chapter 2 Matrices 22 Inverses Let (S) a 11 x 1 + a 12 x 2 + +a 1n x n = b
More informationSPRING OF 2008 D. DETERMINANTS
18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
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 informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -
More informationProperties of the Determinant Function
Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant:
More informationn n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full
n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x
More informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More informationRow Space and Column Space of a Matrix
Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.
More informationSection 4.5. Matrix Inverses
Section 4.5 Matrix Inverses The Definition of Inverse Recall: The multiplicative inverse (or reciprocal) of a nonzero number a is the number b such that ab = 1. We define the inverse of a matrix in almost
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
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 informationMATRICES AND MATRIX OPERATIONS
SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationLinear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationIntroduction to Matrices and Linear Systems Ch. 3
Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts
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 information