Chapter 1 Matrices and Systems of Equations
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1 Chapter 1 Matrices and Systems of Equations
2 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. 2. A linear system of m equations in n variables consists of m linear equations each in variables x 1,..., x n.
3 Solution Set: The set of all n-tuples (x 1,..., x n) that satisfy all m equations in (2) is called the solution set. System is called inconsistent if the solution set is empty. System is called consistent if the solution set is nonempty. Equivalent Systems Two m be n systems are equivalent if their solution sets are the same.
4 Coefficient matrix a 11 x a 1n x n = b 1 a 21 x a 2n x n = b 2. a m1 x a mn x n = b m
5 a 11 a a 1n a 21. a a 2n. a m1 a m2 a mn a 11 a a 1n b 1 a 21. a a 2n. b 2. a m1 a m2 a mn b m
6 If A is an m by n and B is m by r then [A B] denotes the augments m by n + r matrix. Elementary Row Operations 1. Interchange two rows. 2. Multiply a row by a nonzero real number. 3. Replace row r with r+cr where r is a row and c is a number.
7 Pivot element pivotal row = row used to eliminate nonzero entries in the current step. pivot element = nonzero element in the pivotal row which corresponds to the variable which is eliminated from other rows in the current step.
8 Gaussian Elimination The process of using row operations to transform a system to system which is in the row echelon form. Row Echelon Form A matrix is said to be in row echelon form if 1. the first nonzero entry in each row is one; 2. assuming row k contains nonzero entries the number of leading zeros in row k + 1 is greater than the number of leading zeros in row k;
9 3. rows without nonzero entries are below rows with nonzero entries
10 Gauss-Joran Reduction The process of using row operations to transform a system to system which is in the reduced row echelon form. Johann Carl Friedrich Gauss, Reduced Row Echelon Form A matrix is said to be in reduced row echelon form if
11 1. it is in echelon form and 2. the first nonzero entry in a row is the only nonzero entry in its column
12 m by n linear system If m > n then we say the system is overdetermined. If m < n then we say the system is underdetermined. Lead variables = variable that correspond to leading nonzero entries. Free variables = non-leading variables.
13 How can we say from the row echelon form if the system is inconsistent? It contains the row ( a) where a 0. Homogenous systems Systems in which for every i = 1,..., m, b i = 0. Remark. Note that such a system is consistent as it has a trivial solution.
14 Theorem 1 If m < n then a homogenous system with m equations in n variables has a nontrivial solution.
15 Example 1 (Photosynthesis) The chemical equation: carbon dioxide + water + light energy oxygen + glucose. x 1 CO 2 + x 2 H 2 O x 3 O 2 + x 4 C 6 H 12 O 6 Carbon: x 1 = 6x 4 Oxygen: 2x 1 + x 2 = 2x 3 + 6x 4 Hydrogen: 2x 2 = 12x 4
16 This is a homogenous system with four variables and three equations and so it has a nontrivial solution.
17 Example 2 (Economic Model for Exchange of Goods) Design a fair monetary system for a given bartering system. 1/2 1/4 1/3 1/2 1/4 1/4 1/3 1/3 1/4 We have 1 2 x x x 3 = x 1
18 1 4 x x x 3 = x x x x 3 = x 3 which is homogenous 3 by 3 which turns out to have a nontrivial solution.
19 Matrix Algebra A 1 by n matrix is called a (row) vector. An n be 1 matrix is called a (column) vector. For matrix A = [a ij ] we use a(i,:) to denote the ith row in A and a(:, j) or a j for the jth column. Two m n matrices A = [a ij ] and B = [b ij ] are equal if for every i = 1,..., m, j = 1,..., n, a ij = b ij.
20 Operations Scalar Multiplication. Let A = [a ij ] be an m n matrix and let α be a number. Then αa is the m n matrix whose (i, j)th entry is αa ij. The sum. Let A = [a ij ] and B = [b ij ] be two m n matrices, the sum A + B is the m n matrix whose (i, j)th entry is a ij + b ij. The scalar product of vectors. Let a = [a i ] be a 1 n matrix and b = [b i ] be an n 1 vector then ab = n i=1 a i b i.
21 The product. Let A = [a ij ] be an m n matrix and let B = [b ij ] be an n r matrix then the product AB = [c ij ] is the m r matrix whose (i, j)th entry c ij = n k=1 a ik b kj. Other products. Hadamard product, Kronecker product. The transpose. The transpose of an m n matrix A = [a ij ] is the n m matrix A T = [b ij ] where b ij = a ji.
22 Systems of Linear Equations System of linear equations: a 11 x a 1n x n = b 1 a 21 x a 2n x n = b 2. can be written as a m1 x a mn x n = b m Ax = b
23 where A = a 11 a a 1n a 21. a a 2n. a m1 a m2 a mn x = x 1 x 2. x n b = b 1 b 2. b m. Definition 1 Let a 1,a 2, a n be vectors in R m and let c 1, c 2,..., c n R. Then c 1 a 1 + c 2 a c n a n R m
24 is said to be a linear combination of vectors a 1,a 2, a n. Theorem 2 A linear system Ax = b is consistent if and only if b can be written as a linear combination of column vectors of A. Proof. Note that = x 1 Ax = a 11 a 21. a m1 a 11 x 1 + a 12 x a 1n x n a 21 x 1 + a 22 x a 2n x n. a m1 x 1 + a m2 x a mn x n + x 2 a 12 a 22. a m2 + + x n = a 1n a 2n. a mn =
25 = x 1 a x n a n. Theorem 3 Let A, B, C be matrices and let α, β be real numbers. Then A + B = B + A, A + (B + C) = (A + B) + C A(BC) = (AB)C A(B + C) = AB + AC (A + B)C = AC + BC
26 (αβ)a = α(βa),(α + β)a = αa + βa α(ab) = (αa)b, α(a + B) = αa + αb. If A is a square matrix, A k = AA A. Then we can define other functions on matrices. For example, e A = k=0 A k /k!.
27 The 0 matrix, 0 = The identity matrix, n n, I = Definition 2 An n n matrix is called nonsingular (or invertible) if there is a matrix B such that AB = BA = I. Then B is unique and is called the inverse of A.
28 Definition 3 An n n matrix is called singular if it is not invertible. Theorem 4 If A, B are n n nonsingular matrices, then AB is nonsingular and (AB) 1 = B 1 A 1. Theorem 5 1. (A T ) T = A, (αa) T = αa T, (A+B) T = A T +B T. 2. (AB) T = B T A T. Proof.
29 The (i, j)th entry in (AB) T is the (j, i)th entry in AB which is n k=1 a jk b ki. The (j, i)th entry in B T A T is n k=1 b ki a jk. Definition 4 An n n matrix is called symmetric if A T = A. Example 3 Let A be a nonsingular matrix. Then A 1 is nonsingular with (A 1 ) 1 = A and A T is nonsingular with (A T ) 1 =
30 (A 1 ) T. Example 4 (Demographics of the Loggerhead turtle) Divide the life cycle into stages.
31 Assumptions: (1) population size at each stage of a cycle depends on the female population only (2) survival of an individual female depends only on the stage in the cycle (not the age). Four-stage model: (1) Hatchlings (0-1 year), (Annual Survivorship: 0.67), (Eggs laid per year: 0).
32 (2) Juveniles (1-21), (Survivorship: 0.74), (Eggs: 0). (3) Novice breeders (22), (Survivorship: 0.81), (Eggs: 127). (4) Mature breeders (23-54), (Survivorship: 0.81), (Eggs: 79). Let s i, d i be survivorship, and the duration of stage i. Let p i be the proportion of turtles that are stage i and stay in stage i during one year. Let q i be the proportion of turtles that are stage i and will survive and move to stage i + 1. Then p i = 1 sd i 1 i 1 s d i i s i
33 q i = sd i i (1 s i) 1 s d. i i Let e i be the average number of eggs laid by a turtle at stage i. Leslie matrix is L = p 1 e 2 e 3 e 4 q 1 p q 2 p q 3 p 4 = and if the initial population at stage i is x i then the size of population after k years is L k x where x = [x 1,..., x 4 ] T. For example when k = 1, The number of new turtles: p 1 x 1 + e 2 x 2 + e 3 x 3 + e 4 x 4.
34 The number of turtles at stage 3: q 2 x 2 + p 3 x 3. Example 5 A graph consists of a set of vertices and edges, unordered pairs of vertices. Let G = (V, E) be a graph on n vertices with V = {v 1,..., v n }. The adjacency matrix of G is the n n matrix [a ij ] with a ij = 1 if {v i, v j } as an edge and a ij = 0 otherwise. Then A k = [a (k) ij ] where a(k) ij is equal to the number of walks of length k from v i to v j. Indeed, the number of walks of length k from v i to v j is n l=1 and so we are done by induction. a (k 1) il a lj
35 Elementary Matrices Three types: Type 1: Interchange two rows in the identity matrix, I. Type 2: Multiply a row of I by a nonzero number. Type 3: Add a multiple of one row of I to another. Consider Ax = b where A is n n and let E i be an n n matrix of type 4i. Then E i A corresponds to the ith elementary row operation.
36 Theorem 6 Let E be an elementary matrix. Then E is nonsingular and E 1 is an elementary matrix. Proof. Type 1 is the inverse of itself. Type 2 is trivial. If there is m in the ith row and jth column of E then consider the matrix with m in the ith row and jth column. Definition 5 A matrix B is row equivalent to A if there is a finite sequence of elementary matrices E 1,..., E l such that B = E l E 1 A.
37 Theorem 7 Let A be an n n matrix. The following conditions are equivalent. 1. A is nonsingular. 2. Ab = 0 has only one solution. 3. A is row equivalent to I. If E l E l 1 E 1 A = I
38 then E l E l 1 E 1 I = A 1 Triangular matrices: A matrix A = [a ij ] is called strictly triangular if in ith row the first i 1 entries are 0 but a ii 0. An n n matrix A = [a ij ] is called upper triangular (lower triangular) if a ij = 0 for i > j ( a ij = 0 for i < j). An n n matrix A = [a ij ] is called diagonal if a ij = 0 when i j.
39 Block matrices: We can partition a matrix into blocks i.e. A = where each A ij is a matrix. If B = A 11 A A 1l A 21 A A 2l.... A k1 A k2 A kl B 11 B B 1r B 21 B B 2r.... B l1 B l2 B lr
40 and blocks can be multiplied then AB = where C 11 C C 1r C 21 C C 2r.... C k1 C k2 C ij = l t=1 A it B tj. C kr Let x,y are two column vectors. Then x T y is called the inner product. Let x,y are two column vectors. Then x y T is called the
41 outer product of vectors. If A is n r and B is m r then AB T is defined and is called the outer product expansion of A and B. We have AB T = r j=1 a j b T j where a j b T j is the outer product of the jth column of A with the jth column of B.
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