CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS 11 Matrices and Vectors In many occasions, we can arrange a number of values of interest into an rectangular array For example: Example We can represent the information on July sales more simply as 1 15 0 8 5 5
Definitions elements of R, the set of real numbers A matrix is a rectangular array of scalars If the matrix has m rows and n columns, we say that the size of the matrix is m by n, writtenm n The matrix is called square if m = n The scalar in the ith row and jth column is called the (i, j)-entry of the matrix A = Notation: a 11 a 1n a m1 a mn 5 =[a ij ] M m n Example: We use M m n to denote the set that contains all matrices whose sizes are m n
Equality of matrices equal: We say that two matrices A and B are equal if they have the same size and have equal corresponding entries Let A, B M m n Then A = B, a ij = b ij, 8i =1,, m, j =1,,n Example
Submatrices submatrix: A submatrix is obtained by deleting from a matrix entire rows and/or columns For example, E = 15 0 B = 8 15 0 5 5 is a submatrix of 5
Matrix addition Sum of matrices Definition Example 1 5 5 + 1 1 1 1 1 5 = 5 8 5 5
Scalar multiplication Definition Example 1 1 5 5 = 9 1 15 5
Zero matrices zero matrix: matrix with all zero entries, denoted by O (any size) or O m n For example, a -by- zero matrix can be denoted O = 0 0 0 0 0 0 Property A = O + A for all A Property 0 A = O for all A
Question Let A = 1 M and B = 1 5 5 M Then both 0 A and 0 B can be denoted by O, that is, Can we conclude that 0 A = O, and 0 B = O 0 A =0 B? 8
Matrix Subtraction Definition Example 1 1 5 5 =( 1) 1 1 5 5 = 1 1 5 5 9
Question For any m n matrices A and B (ie, 8A, B M m n ), will A + B = B + A always be true? Question For any m n matrices A and B (ie, 8A, B M m n ) and any real number s (ie, 8s R), will s(a + B) =sa + sb always be true? 10
Question For any m n matrices A and B (ie, 8A, B M m n ), will A + B = B + A always be true? Answer: Yes! It is always true Proof: 11
Theorem 11 (Properties of Matrix Addition and Scalar Multiplication) Proof: All proofs can follow from basic arithmetic laws in R and previous definitions Please do all of them yourself (homework) By (b), sum of multiple matrices are written as A + B + + M 1
Transpose Definition Property C M m n ) C T M n m Example C = 18 1 9 5 8 5 ) C T = 18 5 9 1 8 Question Question 1 Is C = C T always wrong? Is 8A, B M m n, (A + B) T = A T + B T always true?
Theorem 1 (Properties of the Transpose) Proof: 1
Vectors A row vector is a matrix with one row A column vector is a matrix with one column 1 1 5 or 1 T The term vector can refer to either a row vector or a column vector (Important) In this course, the term vector always refers to a column vector unless being explicitly mentioned otherwise 15
Vectors R n :We denote the set of all column vectors with n entries by R n In other words, R n = M n 1 components: the entries of a vector Let v R n and assume v = v 1 v v n Then the ith component of v refers to v i 5 1
Vector Addition and Scalar Multiplication Definitions of vector addition and scalar multiplication of vectors follow those for matrices 0 is the zero vector (any size), and u + 0 = u, 0u = 0 for all u R n A matrix is often regarded as a stack of row vectors or a cross list of column vectors For any C M m n, we can write C = c 1 c j c n where c j = c 1j c j c mj 5 1
Geometrical Interpretations Vectors for geometry in R in R vector addition scalar multiplication for a vector 18
Section 11 (Review) Matrix Rows and columns Size (m-by-n) Square matrix (i,j)-entry Matrix Equality, Addition, Zero Matrix Scalar multiplication, subtraction Vector Row vectors, column vectors components 19
1 Linear Combinations, Matrix-Vector Products, and Special Matrices Definition Example: 8 = 1 1 1 + 1 +1 1 Given the coefficients ({-,,1}), it is easy to compute the combination ([ 8] T ), but the inverse problem is harder 0 Example: 1 = x 1 + x 1 = x1 x + = x 1 1x x1 +x x 1 + x To determine x 1 and x, we must solve a system of linear equations, which has a unique solution [x 1 x ] T = [-1 ] T in this case
Geometrical view point: manage to form a parallelogram Example: to determine if [- -] T is a linear combination of [ ] T and [ 1] T, we must solve x 1 +x = x 1 + x = which has infinitely many solutions, as the geometry suggests 1
Example: to determine if [ ] T is a linear combination of [ ] T and [ ] T, must solve x 1 +x = x 1 +x = which has no solutions, as the geometry suggests
If u and v are any nonparallel vectors in R, then every vector in R is a linear combination of u and v (unique linear combination) algebraically, this means that u and v are nonzero vectors, and u cv What is the condition in R? in R n?
The standard vectors of R n are defined as Obviously, every vector in R n may be uniquely linearly combined by these standard vectors e 1 = 1 0 0 5, e = 0 1 0 5,, e n = 0 0 1 5 Standard vectors
Matrix-Vector Product Definition Note that we can write: Example: Av = 1 5 Av = Let A = 1 5, v = 8 5 5 = 5 +8 8 1 5 a 1 a a n ThenAv =? 5 = 1 5 5 v 1 v v n 5 + 5 1 8 5 = 5 8 5 5 Property: A0 = 0 and Ov = 0 for any A and v
Let A M and v R Then v a11 a Av = 1 a 1 1 v a 1 a a 5 a11 = v 1 a v 1 + v a1 a + v a1 a = a11 v 1 + a 1 v + a 1 v a 1 v 1 + a v + a v More generally, when A M m n and v R n Then a 11 a 1 a 1n v 1 a 11 a 1 a a n v a 1 Av = a m1 a m a mn 5 v n 5 = v 1 = a m1 The ith component of Av is 5 + v a 1 a a m a 11 v 1 + a 1 v + + a 1n v n a 1 v 1 + a v + + a n v n 5 + + v a m1 v 1 + a m v + + a mn v n 5 ai1 a i a 1n a 1n a n a mn v 1 v 5 5 v n
Identity Matrix Definition Example: I = 1 0 0 0 1 0 0 0 1 5 Sometimes I n is simply written as I (any size) Property: I n v = v for any v R n
Stochastic Matrix Definition An n n matrix A M n n is called a stochastic matrix if all entries of A are nonnegative and the sum of all entries in each column is unity Example: A = 085 00 015 09 is a stochastic matrix 8
Example: stochastic matrix To City Suburbs From City Suburbs 85 0 = A 15 9 probability matrix of a sample person s residence movement p = 500 00 : current population of the city and suburbs This year City 500 thousand Suburbs 00 thousand 85% 15% % 9% Next year 085 x 500 + 00 * 00 = City 85 0 500 Ap = 15 9 00 015 x 500 + 09 x 00 = 5 Suburbs : population distribution in the next year 9 A(Ap) : population distribution in the year following the next
Example: rotation matrix A = A x y cos sin = sin cos cos sin sin cos x y P 0 = x 0 y 0 P = x y = x cos y sin x sin + y cos = = x x 0 y 0 cos sin + y sin cos 0
Question Is the statement (A + B)u = Au + Bu, 8A, B M m n, u R n always true? Question Let A M m n and e j be the jth standard vector in R n Then what is Ae j? 1
Theorem 1 (Properties of Matrix-Vector Products) Proof for (e): If B A, then (B - A)e j 0, ie, Be j Ae j, for some j
Problems for practice (11~1) Section 11: Problems 1, 5,, 11, 1, 19, 5,, 9, 1,, 9, 1,, 5, 51, 5, 55 Section 1: Problems, 5, 8, 9, 15,,,, 8-8