Brief Introduction to Vectors and Matrices


 Amice Powers
 2 years ago
 Views:
Transcription
1 CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vectorvalued function, and linear independency of a group of vectors and vectorvalued functions. 1. Vectors and Matrices A matrix is a group of numbers(elements) that are arranged in rows and columns. In general, an m n matrix is a rectangular array of mn numbers (or elements) arranged in m rows and n columns. If m n the matrix is called a square matrix. For example a 2 2 matrix is a11 a 12 and an 3 3 matrix is a 21 a 22 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Generally, we use bold phase letter, like A, to denote a matrix, and lower case letters with subscripts, like a ij, to denote element of a matrix. Here a ij would be the element at i th row and j th column. So a 11 is an element at 1 st row and column. Sometime we use the abbreviation A (a ij ) for a matrix with elements a ij Special matrices. 0 denotes the zero matrix whose elements are all zeroes. So 2 2 and 3 3 zero matrices are 0 0 and Another special matrix is the identity matrix, denoted by I, a identity matrix is an matrix whose main diagonal elements are 1, and all 1
2 2 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES other elements are 0. So 2 2 and 3 3 zero matrices are 1 0 and A vector is a matrix with one row or one column. In this chapter, a vector is always a matrix with one column as x1 x 2 for a twodimensional vector and x 1 x 2 x 3 for a three dimensional vector. Here the element has only one index that denotes the row position (Sometimes we use different variable to denote number in different position such as using x y for a 2dimensional vector). denote a vector. We use bold lower case, such as v, to 1.2. Operations on Matrices. Arrange number in rectangular fashion, as a matrix, itself is not something terribly interesting. The most important advantage from that kind arrangement is that we can define matrix addition, multiplication, and scalar multiplication. Definition 1.1. (i) Equality: Two matrix A (a ij ) and B (b ij ) are equal if corresponding elements are equal, i.e. a ij b ij. (ii) Addition: If A (a ij ) and B (b ij ) and the sum of Aand B is A + B (c ij ) a ij + b ij. (iii) Scalar Product: If A (a ij ) is matrix and k is number(scalar), the ka (ka ij ) is product of k and A. From the above definition, we see that, to multiply a matrix by a number k, we simply multiply each of its entries by k; to add two matrices we just add their corresponding entries; A B A+( 1)B. Example 1.1. Let A
3 and 1. VECTORS AND MATRICES 3 B find (a) A + B, (b) 3A, (c) 4A B. Solution (a) (b) (c) 4A B, A + B ( 4) A The following fact lists all properties of matrix addition and scalar product. Theorem 1.1. Let A, bb, and C be matrices. Let a, b be scalars (numbers). We have (1) A A A, A A 0; (2) A + B B + A (commutativity); (3) A + (B + C) (A + B) + C, (ab)a a(ba) (associativity); (4) a(a+b) aa+ab, (a+b)a aa+ba (distributivity) When we have a row vector and a column vector with the same number of elements, we can define the dot product as Definition 1.2. Dot Product: in 2dimension: Let x y1 x 1 x 2 and y, the y 2 dot product of x and y is, x y x 1 y 1 + x 2 y 2
4 4 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES in 3dimension: Let x x 1 x 2 x 3 and y the dot product of x and y is, y 1 y 2 y 3, x y x 1 y 1 + x 2 y 2 + x 3 y 3 Definition 1.3. Matrix product Let A (a ij ) and B (b ij ), if the number of columns of A is the same as number of rows of B, then the product of A and B is given by AB (c ij ) where c ij is dot product of i th row of A with j th column of B. and Example 1.2. Let find AB A B , Solution AB (0) + 3 (3) ( 4) ( 1) (0) ( 4) Notice, the first element of AB is 2 (0) + 3 (3) which is the dot product of first row of A, and first column of B, 3 The following fact gives properties of matrix product, Theorem 1.2. Let A, B, C be three matrices and r be a scalar, we have A(BC) (AB)C, r(ab) A(rB) (associativity) A(B + C) AB + AC (distributivity) Notice, in general AB BA, that is for most of the times, AB is not equal to BA. Using the matrix notation and matrix product, we can write the following system of equations { ax1 + bx 2 y 1 cx 1 + dx 2 y 2
5 as Ax y with x x1 x 2, and y 1. VECTORS AND MATRICES 5 a b A c d. y 2 y1 Definition 1.4. A square (ex. 2 2 or 3 3) matrix A is invertible if there is a matrix A 1 such that AA 1 A 1 A I. Theorem 1.3. Let, a b A c d be a 2 2 matrix, if A is invertible, we have A 1 1 d b ad bc c d So if A is invertible, to solve Ax y, we need to simply multiply both sides with A 1, that is x x A 1 y. Example 1.3. Solve the system of equation { 3x1 4x 2 2 2x 1 + 5x 2 7 Solution x1 equation is The equation can be rewrite Ax y with 3 4 A, x 2, and y Now the inverse of A is so the solution is A 1 x A 1 y 1 7. So in matrix form the system of x1 2 x (5) ( 2)( 4) ,
6 6 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES x Example 1.4. Solve the system of equation 3x 1 4x 2 + 5x 3 2 2x 1 + 5x 2 7 x 1 5x 2 + 8x 3 1 Solution x 1 x 2 x 3 The equation can be rewrite Ax y with A , 1 5 8, and y equation, Ax y, is So in matrix form the system of x 1 x 2 x It is a little harder to compute the inverse of a 3 3 matrix, we will use Mathcad to solve the equation. Here is how to do it, Type A:CtrlM at a blank area to bring up the matrix definition screen, put 3 in the both input boxes and click OK, you will get a 3 3 matrix place holder like A :. Fill the entries of A in the corresponding position, using Tab key to navigate among the place holders(or just click each one). Type b:ctrlm in another blank area, the matrix definition screen is up again. This time put 3 in the number of row box, and 1 in the number of column box and click OK. You will get b: put the values of y in the corresponding position. Type A^1 *b you will get the solution, which is,
7 1. VECTORS AND MATRICES 7 Notice, by default, Mathcad will display the results as decimal, you can double click on the result vector to change it to fraction, after you double click the result you will have a popup menu such as Figure 1. Format Result The next example shows how we can determine the unknown constants typically found in the initial value problems of system of differential equations. Example 1.5. Let x 1 (t) C 1 e t + C 2 e 2t and x 2 2C 1 e t C 2 e 2t. If x 1 (0) 2, x 2 (0) 3, find C 1 and C 2 Solution From x 1 (t) C 1 e t + C 2 e 2t, set t 0 we have x 1 (0) C 1 e 0 + C 2 e 2(0) C 1 + C 2. Similarly, x 2 (t) 2C 1 e t C 2 e 2t, gives x 2 (0) 2C 1 e ( 0) C 2 e 2(0) 2C 1 C 2. Together with x 1 (0) 2, x 2 (0) 3 we have the following system of equations, { C1 + C 2 1 2C 1 C 2 3 Rewrite the equation in matrix form 1 1 C C 2 3 and using Mathcad we find the solution is 4 C1 3 C 2 1 3,
8 8 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES So x 1 (t) 4 3 et 1 3 e 2t and x et e 2t. They are solution of the following system of differential equations, { x 1 (t) x 1(t) + x 2 (t) x 2 (t) 2x 1 a b 1.3. Eigenvalues and Eigenvectors. If A the de c d termined of A is defined as A a b c d ad bc. For a 3 3 matrix a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 we can compute the matrix as a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 11 a 22 a 23 a 32 a 33 a 12 a 21 a 23 a 31 a 33 +a 13 a 21 a 22 a 31 a 32 In Mathcad, type the vertical bar to bring up the absolute evaluator, put the matrix in the place holder and press to compute the determinant. The following screen shot shows an example,. Figure 2. Compute determinant in Mathcad The concepts of eigenvalue and eigenvector play an important role in find solutions to system of differential equations. Definition 1.5. We say λ is an eigenvalue of a matrix A (2 2 or 3 3) if the determinant A λi 0. An nonzero vector v is an eigenvector associated with λ if Remark 1.1. Av λv.
9 1. VECTORS AND MATRICES 9  The above definition of eigenvector and eigenvalue is valid for any square matrix with n rows and columns.  p(λ) A λi is a polynomial of degree n for n n matrix A, which is called the characteristic polynomial of A.  If we view A as an transform that maps a vector x to Ax, an eigenvector v defines a straight line passing origin that is invariant under A.  If v is an eigenvector then for and number s 0, sv is also an eigenvector. This is especially useful when using Mathcad to get eigenvectors, the result of Mathcad might look bad, you might need to remove the common factor of the component of the vector to make it better. Computing eigenvalues and eigenvectors of a given matrix is quite tedious, Mathcad provides two functions eigenvals() and eigenvecs() to compute eigenvalues and eigenvectors of a matrix. In Mathcad, eigenvecs(m) Returns a matrix containing the eigenvectors. The nth column of the matrix returned is an eigenvector corresponding to the nth eigenvalue returned by eigenvals. The results of these functions by default is in decimal, you can change it by using simplify key word as shown in the following diagram. (a) Find eigenvalue (b) Find eigenvector Figure 3. Compute eigenvalue and eigenvector in Mathcad Notice, in the diagram, the eigenvalues are listed as vector and the eigenvectors are listed in a matrix 1+ 3 (8+2 3) (8+2 3) 1 2 ( 31) (82 3) (82 3) 1 2, 3 3
10 10 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES each column represents a eigenvector. Since multiplying an eigenvector by a nonzero constant you still get an eigenvector, so we can simplify the eigenvectors as v 1, and v Here is how to use Mathcad, Define the matrix by type A:CtrlM and specify the row and column number, fill the entries. type eigenvals(, you will get eigenvals( ) and in the place holder type A. Click at end of the eigenvals(a) and press ShiftCtrl., you will get eigenvals(a). In the place holder type in key word simple. And click any area outside the box to get result. Using the same procedure for find eigenvector using eigenvecs() function. 2. Vectorvalued functions A vectorvalued function over a, b is a function whose value is a vector or matrix. For example the following functions are vectorvalued functions, t Example 2.1. (1) v(t) t 2 (2) x 1 t 2 (3) A(t) e t 1 t 3 4t sin(t) 2.1. Arithmetics of vectorvalued function. To add two vectorvalued function is to add their corresponding components. To multiply a vectorvalued function by a scalar function to to multiply each entry by the scalar function. To multiply a vector(matrix) valued function to another vectorvalued function is same as multiply a matrix with a vector. The following example illustrate how to add/subtract two vectorvalued functions and how to multiply a vectorvalued function by a scalar function and how to apply a vectorvalued function that is matrix to a vector value function.
11 2. VECTORVALUED FUNCTIONS 11 Example 2.2. Suppose v(t) A(t) t t 2 t sin(t) 1 t3 4t (a) Find v(t) + x(t); (b) Let f(t) e t, find f(t)x(t); (c) Find A(t)x(t), x 1 t 2 e t., and Solution (a) v(t) + x(t) (b) (c) t t 2 t t 2 e t f(t)x(t) e t 1 t 2 e t et t 2 e t e 2t A(t)x(t) 1 t3 4t sin(t) t2 (t 3 4t + 5) + e t 2t 2 + e t sin(t) 2 + sin(t) t + 1 2t 2 t e t ; 1 t 2 e t ; 2.2. derivative and integrations of vectorvalued functions. A vectorvalued function is continuous if each of its entries are continuous. A vectorvalued function is differentiable if each of its entries are differentiable. If v(t) is an vectorvalued function, then the derivative dv(t) dt v (t) of v(t) is a vectorvalued function whose entries are the derivative of corresponding entries of v(t). That is to find derivative of a vectorvalued function we just need to find derivative of each of its component.
12 12 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES The antiderivative v(t) dt of an vectorvalued function v(t) is a vectorvalued function whose entries are the antiderivative of corresponding entries of v(t). 3t Example 2.3. Find derivative of x(t) 2 5 sin(t) Solution x (t) dx(t) dt d dt 3t 2 5 sin(t) d dt (3t2 5) d (sin(t)) dt Example 2.4. Find antiderivative of x(t) 3t 2 5 sin(t) 6t cos(t) Solution 3t 2 5 x(t) dt sin(t) t 3 5t + C 1 cos(t) + C 2 (3t dt 2 5) dt sin(t) dt C1 + C 2 t 3 5t cos(t) Theorem 2.1. Suppose v(t), x(t), A(t) are differentiable vectorvalued functions (A(t) is matrix), and f(t) is differentiable scalar function. We have, (1) Sum and Difference rule:  v(t) ± x(t) v (t) ± x (t),  v(t) ± x(t) dt v(t) dt ± x(t) dt. (2) Product rule:  f(t)v(t) f (t)v(t) + f(t)v (t),  A(t)x(t) A (t)x(t) + A(t)x (t), Using Mathcad to find derivative or antiderivative of a vectorvalued function using Mathcad, you need to find derivative or antiderivative component wise as shown in the following screen shot,
13 2. VECTORVALUED FUNCTIONS 13 Figure 4. Differentiate and integrate vectorvalued function
14 14 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES Notice:  Press Shift/to get the derivative operator and press CtrlI to get the antiderivative operator.  To get dx(t)simplif y you type dx(t) and press ShiftCtrl. and type the key word simplify in the place holder before.  To execute symbolically ( operator), just press Ctrl. 3. Linearly independency 3.1. Linearly independency of vectors. Let x 1, x 2,, x n be n vectors, C 1, C 2,, C n are n scalars(numbers), the expression C 1 x 1 + C 2 x C n x n is called a linear combination of vectors x 1, x 2,, x n. if Definition 3.1. n vectors x 1, x 2,, x n is linearly independent C 1 x 1 + C 2 x C n x n 0 leads to C 1 0, C 2 0,, C n 0. A set of vectors are linearly dependent if they are not linearly independent. If 0 is one of x 1, x 2,, x n, then they linearly dependent. Two nonzero vectors x and y are linearly dependent if and only if x sy for some s 0. n nonzero vectors are linearly independent if one can be represented as linear combination of the others. Any three or more 2dimensional vectors (vectors with two entries) are linear dependent. Any four or more 3dimensional(vectors with three entries) vectors are linear dependent. To determine if a given set of vectors are linearly independent, create a matrix so that the row of the matrix are given vectors. Using Mathcad function rref( ) to find the reduced echelon form of the matrix, if the result contains one or more rows that are entirely zero the vectors are linearly dependent, otherwise the vectors are linearly independent.
15 3. LINEARLY INDEPENDENCY 15 Example 3.1. For x , x , and x , we can form a matrix, A apply rref(type rref and in the place holder type A, then press ), rref(a) We see that the vectors are linearly dependent as the last row is entirely zero Linearly independency of functions. We can also define linearly independency for a group of functions over an given interval a, b. Let f 1, f 2,, f n be n functions defined over a, b, C 1, C 2,, C n are n scalars(numbers), the expression, C 1 f 1 + C 2 f C n f n is called a linear combination of functions f 1, f 2,, f n. Definition 3.2. n functions f 1, f 2,, f n is linearly independent over a, bif (1) C 1 f 1 + C 2 f C n f n 0 for all a t b leads to C 1 0, C 2 0,, C n 0. A set of function are linearly dependent if they are not linearly independent. If 0 function is one of f 1, f 2,, f n, then they linearly dependent. Two nonzero functions f(t) and g(t) are linearly dependent over a, b if and only if f(t) sg(t) for a constant s 0 and all a t b, for example, f(t) t and g(t) 4t are linearly dependent but f(t) t and g(t) 4t 2 are not, even f(0) 4g(0) and f(1) 4g(1). There are exists infinite many functions that are linearly independent. For example the set {1, t, t 2, t 3,, t n, } is a linearly independent set.
16 16 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES Here are some sets of linearly independent functions that we encounter in solving a system of differential equations, assume k 1, k 2,, k n are different numbers,  {t k 1, t k 2,, t k n }.  {e k 1t, e k 1t,, e k 1t }.  {sin(k 1 t), sin(k 2 t),, sin(k n t)}.  {cos(k 1 t), cos(k 2 t),, cos(k n t)}.  The mixing of above sets.  For each above set, when multiplying each element by an common nonzero factor, we get another linearly independent set. The following screen shot displays a heuristic Mathcad function that tries to determine if a given set of functions are linearly independent. Figure 5. Calculus tool bar One warning, the result of the program is not very reliable, the user should check the result manually to confirm the result. To manually check if an set of functions are linearly independent on a, b, one need to show that the only solutions are C 1 0, C 2 0,, C n 0. if equation (1) holds for all t in a, b, which requires strong algebraic skill. One method is to choose n different numbers {t 1, t 2,, t n } from a, b and using the functions to create an matrix, the compute the determinant of the matrix A (f i (t j )), if the determinant is not zero, the functions are linearly independent, but if the determinant is zero, it is inconclusive(most likely are linearly dependent). Example 3.2. Determine if f 1 (t) t 2 2t + 3, f 2 (t) 2t 2 5t 6, and f 3 (t) 5t 2 11t + 4 are linearly independent.
17 3. LINEARLY INDEPENDENCY 17 Solution Choose t 1 1, t 2 0, and t 3 1, f 1(t 1 ) f 1 (t 2 ) f 1 (t 3 ) f 2 (t 1 ) f 2 (t 2 ) f 2 (t 3 ) f 3 (t 1 ) f 3 (t 2 ) f 3 (t 3 ) Compute the determinant, , 2 so the functions are linearly independent. Project At beginning you should enter: Project title, your name, ss#, and due date in the following format Project One: Define and Graph Functions John Doe SS# Due: Mon. Nov. 23rd, 2003 You should format the text region so that the color of text is different than math expression. You can choose color for text from Format >Style select normal and click modify, then change the settings for font. You can do this for headings etc. (1) Independent of functions as vectors Goal: Familiar your self with the concept of linearly independency. Use the Mathcad code provided at at the website mzhan/linear to check if given set of functions are linearly independent or not. {sin(x), sin(2x), sin(3x)} {t 2, 2t 2 2t + 4, 3t, 6} {e t, te t, t 3 e t } {e 2t, e t, e 3t } Using algebraic arguments or reasoning to verify the conclusion of the Mathcad code.
18 18 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES (2) Condition Number In solving Ax b, one number is very important, it is called the condition number, which can be defined as C(A) s, where λ s is the eigenvalue with smallest l absolute value and lambda l is the eigenvalue with largest absolute value, if C(A) is too large or too small, a little change in b will result in a large in the solution x. We say the system Ax b is not stable. Now if A Find all eigenvalues, all eigenvectors, and C(A). Find solution of Ax b if b Change b a little to b 1 1 we get different solution, 1.1 which component of the new solution change most? The change of the third component if 10% what is the percentage change of the most changed component? Note: Our definition of condition number is not accurate, the true definition is C(A) where is a 1 A A 1 given norm (metric). Mathcad provides three functions cond1(a), cond2(a) and condi(a) in compute condition number for A in different metric
System of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More informationMATH 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 mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationMATH 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
More informationIntroduction to Mathcad
CHAPTER 1 Introduction to Mathcad Mathcad is a product of MathSoft inc. The Mathcad can help us to calculate, graph, and communicate technical ideas. It lets us work with mathematical expressions using
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationMATRIX 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 +
More information2.1: 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
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More information1 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
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationSimilarity 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
More informationMATRIX 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
More informationMATH 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
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More information9 Matrices, determinants, inverse matrix, Cramer s Rule
AAC  Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationNON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationMATH 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
More informationDiagonalisation. 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
More informationSergei 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
More information1 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
More informationLinear 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
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationr (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)
Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system
More informationSection 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
More informationOrthogonal 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
More informationB 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.
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationChapter 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
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationLinear 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
More informationa 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.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationMath 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)(
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationMath 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
More informationTopic 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
More informationMATH 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
More informationFurther Maths Matrix Summary
Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationLecture 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
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationINTRODUCTORY 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
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More information2.5 Elementary Row Operations and the Determinant
2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)
More informationNOTES 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
More informationLinear Algebra and TI 89
Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMath 2331 Linear Algebra
2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University
More informationChapter 8. Matrices II: inverses. 8.1 What is an inverse?
Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we
More information1. 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
More informationSolving 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
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More informationLecture 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
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More information1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams
In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th
More informationChapter 4  Systems of Equations and Inequalities
Math 233  Spring 2009 Chapter 4  Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to
More informationSolutions 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
More informationAu = = = 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
More informationMathQuest: 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
More informationMatrix Algebra and Applications
Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters
More informationSection 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,
More informationName: 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
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationC 1 x(t) = e ta C = e C n. 2! A2 + t3
Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix
More informationNotes on Matrix Multiplication and the Transitive Closure
ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationMatrix Methods for Linear Systems of Differential Equations
Matrix Methods for Linear Systems of Differential Equations We now present an application of matrix methods to linear systems of differential equations. We shall follow the development given in Chapter
More informationA Brief Primer on Matrix Algebra
A Brief Primer on Matrix Algebra A matrix is a rectangular array of numbers whose individual entries are called elements. Each horizontal array of elements is called a row, while each vertical array is
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More information5.3 Determinants and Cramer s Rule
290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given
More informationSECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or
More informationHomework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67
Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of
More informationBasic 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
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More information( ) 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
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More information1 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
More information