Brief Introduction to Vectors and Matrices

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, vector-valued function, and linear independency of a group of vectors and vector-valued 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 two-dimensional 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 2-dimensional 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 2-dimension: 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 3-dimension: 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 -( 3-1) (8-2 3) (8-2 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. Vector-valued functions A vector-valued function over a, b is a function whose value is a vector or matrix. For example the following functions are vector-valued 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 vector-valued function. To add two vector-valued function is to add their corresponding components. To multiply a vector-valued 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 vector-valued functions and how to multiply a vector-valued function by a scalar function and how to apply a vector-valued function that is matrix to a vector value function.

11 2. VECTOR-VALUED 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 vector-valued functions. A vector-valued function is continuous if each of its entries are continuous. A vector-valued function is differentiable if each of its entries are differentiable. If v(t) is an vector-valued function, then the derivative dv(t) dt v (t) of v(t) is a vector-valued function whose entries are the derivative of corresponding entries of v(t). That is to find derivative of a vector-valued 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 vector-valued function v(t) is a vector-valued 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. VECTOR-VALUED FUNCTIONS 13 Figure 4. Differentiate and integrate vector-valued 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 2-dimensional vectors (vectors with two entries) are linear dependent. Any four or more 3-dimensional(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