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
linear equation in one unknown x: ax = b. 2
Exactly one of following holds: (1) there is precisely one solution x = b/a, a 0, (2) there are no solutions 0x = b, b 0 (3) there are infinitely many solutions 0x = 0. 3
linear equation in two unknowns x, y: ax + by = α. A solution of the equation is an ordered pair of numbers (x, y). If a = b = 0, and α = 0, then all ordered pairs satisfy the equation. If a = b = 0, and α 0, then no ordered pair satisfies the equation. 4
If a, b, not both 0, then the set of all ordered pairs that satisfy the equation is a straight line (in the x, y-plane).
A system of two linear equations in two unknowns: ax + by = α cx + dy = β Find ordered pairs (x, y) that satisfy both equations simultaneously. 5
Two lines in the plane either (a) have a unique point of intersection (the lines have different slopes), 12 10-4 -3-2 -1 1 2 3-2 8 6 4 2 6
(b) are parallel (the lines have the same slope but, for example, different y-intercepts) 3 2 1-3 -2-1 1 2 3 7
(c) coincide (same slope, same y- intercept). 2 1-3 -2-1 1 2 3 8
That is, there is either a (a) unique solution, (b) no solution, or (c) infinitely many solutions. 9
A system of three linear equations in two unknowns: ax + by = α cx + dy = β ex + fy = γ Find ordered pairs (x, y) that satisfy the three equations simultaneously. 10
There is either a (a) unique solution, (b) no solution, (this is usually what happens) or (c) infinitely many solutions. 11
Example: x + y = 2 2x + y = 2 4x + y = 11 10 5-1 1 2 3 12
A linear equation in three unknowns x, y, z: ax + by + cz = α. A solution of the equation is an ordered triple of numbers (x, y, z). If a = b = c = 0, and α = 0, all ordered triples satisfy the equation. If a = b = c = 0, and α 0, no ordered triple satisfies the equation. 13
If a, b, c, not all 0, then the set of all ordered triples that satisfy the equation is a plane (in 3-space). -5 0 5 20 0-20 -5 0 5
A system of two linear equations in three unknowns a 11 x + a 12 y + a 13 z = b 1 a 21 x + a 22 y + a 23 z = b 2 Either the two planes are parallel (the system has no solutions), 14
Figure 2 0 2 40 20 0 20 40 2 0 2 15
they coincide (infinitely many solutions, a whole plane of solutions), they intersect in a straight line (again, infinitely many solutions.) -5 0 5 20 0-20 -5 0 5 16
A system of three linear equations in three unknowns. a 11 x + a 12 y + a 13 z = b 1 a 21 x + a 22 y + a 23 z = b 2 a 31 x + a 32 y + a 33 z = b 3 The system represents three planes in 3-space. 17
(a) The system has a unique solution; the three planes have a unique point of intersection; (b) The system has infinitely many solutions; the three planes intersect in a line, or the three planes intersect in a plane. (c) The system has no solution; there is no point the lies on all three planes. 18
Systems of Linear Algebraic Equations Example 1: Solve the system x + 3y = 5 2x y = 4 19
Equivalent system x + 3y = 5 y = 2 Solution set: x = 1, y = 2 20
Example 2: Solve the system x + 2y 5z = 1 3x 9y + 21z = 0 x + 6y 11z = 1 21
Equivalent system x + 2y 5z = 1 y 2z = 1 z = 1 Solution set: x = 4, y = 1, z = 1 22
Example 3: Solve the system 3x 4y z = 3 2x 3y + z = 1 x 2y + 3z = 2 23
Equivalent system x 2y + 3z = 2 y 5z = 3 0z = 1 The system has no solution. 24
The Elementary Operations The operations that produce equivalent systems are called elementary operations. 1. Multiply an equation by a nonzero number. 2. Interchange two equations. 3. Multiply an equation by a number and add it to another equation. 25
Example 4: Solve the system x 1 2x 2 + x 3 x 4 = 2 2x 1 + 5x 2 x 3 + 4x 4 = 1 3x 1 7x 2 + 4x 3 4x 4 = 4 26
Equivalent system x 1 2x 2 + x 3 x 4 = 2 x 2 + x 3 + 2x 4 = 3 x 3 + 1 2 x 4 = 1 2 27
Solution set: x 1 = 13 2 3 2 a, x 2 = 5 2 3 2 a, x 3 = 1 2 1 2 a, x 4 = a, a any real number. 28
Terms A matrix is a rectangular array of numbers. A matrix with m rows and n columns is an m n matrix. Matrix representation of a system of linear equations 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..... a m1 x 1 + a m2 x 2 + + a mn x n = b m 29
Augmented matrix and matrix of coefficients: Augmented matrix: a 11 a 12 a 1n b 1 a 21 a 22... a 2n. b 2. a m1 a m2 a mn b m Matrix of coefficients: a 11 a 12 a 1n a 21 a 22.. a 2n. a m1 a 32 a mn 30
Elementary row operations: 1. Interchange row i and row j R i R j. 2. Multiply row i by a nonzero number k kr i R i. 3. Multiply row i by a number k and add the result to row j kr i + R j R j. 31
Examples 1. Solve the system x + 2y 5z = 1 3x 9y + 21z = 0 x + 6y 11z = 1 Augmented matrix: 1 2 5 1 3 9 21 0 1 6 11 1 32
Row reduce 1 2 5 1 3 9 21 0 1 6 11 1 33
1 2 5 1 0 1 2 1 0 0 1 1 Corresponding (equivalent) system of equations: x + 2y 5z = 1 y 2z = 1 z = 1 Solution set: x = 4, y = 1, z = 1 34
2. Solve the system: 3x 4y z = 3 2x 3y + z = 1 x 2y + 3z = 2 Augmented matrix: 3 4 1 3 2 3 1 1 1 2 3 2. 35
Row reduce 3 4 1 3 2 3 1 1 1 2 3 2 36
Equivalent system 1 2 3 2 0 1 5 3 0 0 0 1 Corresponding system of equations: x 2y + 3z = 2 0x + y 5z = 3 0x + 0y + 0z = 1 That is x 2y + 3z = 2 y 5z = 3 0z = 1 Solution set: no solution. 37
3. Solve the system x + y 3z = 1 2x + y 4z = 0 3x + 2y z = 7 Augmented matrix: 1 1 3 1 2 1 4 0 3 2 1 7 38
Row reduce 1 1 3 1 2 1 4 0 3 2 1 7 39
Equivalent system: 1 1 3 1 0 1 2 2 0 0 0 0. Corresponding system of equations: x + y 3z = 1 0x + y 2z = 2 0x + 0y + 0z = 0 or or x + y 3z = 1 y 2z = 2 0z = 0 x + y 3z = 1 y 2z = 2 40
This system has infinitely many solutions given by: x = 1 + a, y = 2 + 2a, z = a, a any real number. 41
Row echelon form: 1. Rows consisting entirely of zeros are at the bottom of the matrix. 2. The first nonzero entry in a nonzero row is a 1. This is called the leading 1. 3. If row i and row i + 1 are nonzero rows, then the leading 1 in row i+1 is to the right of the leading 1 in row i. 42
NOTE: 1. All the entries below a leading 1 are zero. 2. The number of leading 1 s is less than or equal to the number of rows. 3. The number of leading 1 s is less than or equal to the number of columns. 43
Solution method for systems of linear equations: 1. Write the augmented matrix (A b) for the system. 2. Use elementary row operations to transform the augmented matrix to row echelon form. 3. Write the system of equations corresponding to the row echelon form. 44
4. Back substitute to find the solution set. This method is called Gaussian elimination with back substitution. 45
Consistent/Inconsistent systems: A system of linear equations is consistent if it has at least one solution. That is, a system is consistent if it has either a unique solution or infinitely many solutions. A system that has no solutions is inconsistent. 46
Consistent systems: A consistent system is said to be independent if it has a unique solution. A system with infinitely many solutions is called dependent. 47
4. Solve the system of equations 2x 1 + 5x 2 5x 3 7x 4 = 8 x 1 + 2x 2 3x 3 4x 4 = 2 3x 1 6x 2 + 11x 3 + 16x 4 = 0 Augmented matrix: 2 5 5 7 8 1 2 3 4 2 3 6 11 16 0. Transform to row echelon form: 48
Equivalent system: 1 2 3 4 2 0 1 1 1 4 0 0 1 2 3. Corresponding system of equations: x 1 + 2x 2 3x 3 4x 4 = 2 x 2 + x 3 + x 4 = 4 x 3 + 2x 4 = 3 49
Solution set: x 1 = 9 4a, x 2 = 1 + a, x 3 = 3 2a, x 4 = a, a any real number. 50
5. Solve the system of equations x 1 3x 2 + 2x 3 x 4 + 2x 5 = 2 3x 1 9x 2 + 7x 3 x 4 + 3x 5 = 7 2x 1 6x 2 + 7x 3 + 4x 4 5x 5 = 7 Augmented matrix: 1 3 2 1 2 2 3 9 7 1 3 7 2 6 7 4 5 7 Transform to row echelon form: 51
Equivalent system: 1 3 2 1 2 2 0 0 1 2 3 1 0 0 0 0 0 0. Corresponding system of equations: x 1 3x 2 + 2x 3 x 4 + 2x 5 = 2 0x 1 + 0x 2 + x 3 + 2x 4 3x 5 = 1 0x 1 + 0x 2 + 0x 3 + 0x 4 + 0x 5 = 0. which is x 1 3x 2 + 2x 3 x 4 + 2x 5 = 2 x 3 + 2x 4 3x 5 = 1 52
Solution set: x 1 = 3a + 5b 8c, x 2 = a, x 3 = 1 2b + 3c, x 4 = b, x 5 = c, a, b, c arbitrary real numbers 53
6. For what value(s) of k, if any, does the system x + y z = 1 2x + 3y + kz = 3 x + ky + 3z = 2 have: (a) a unique solution? (b) infinitely many solutions? (c) no solution? 54
Augmented matrix: 1 1 1 1 2 3 k 3 1 k 3 2 55
1 1 1 1 0 1 k + 2 1 0 0 (k + 3)(k 2) k 2 (a) Unique solution: k 2, 3. (b) Infinitely many solns: k = 2. (c) No solution: k = 3. 56
If an m n matrix A is reduced to row echelon form, then the number of non-zero rows in its row echelon form is called the rank of A. Equivalently, the rank of a matrix is the number of leading 1 s in its row echelon form. 57
Note: 1. The rank of a matrix is less than or equal to the number of rows. (Obvious) 2. The rank of a matrix is also less than or equal to the number of columns. 58
Consistent/Inconsistent Systems Case 1: If the last nonzero row in the row echelon form of the augmented matrix is (0 0 0 0 1), then that row corresponds to the equation 0x 1 + 0x 2 + 0x 3 + + 0x n = 1, which has no solutions. Therefore, the system has no solutions. 59
Note: In this case, rank of augmented matrix > rank of coefficient matrix
Case 2: If the last nonzero row has the form (0 0 0 1 b), where the 1 is in the k th, k n column, then the row corresponds to the equation 0x 1 + +0x k 1 +x k +( )x k+1 + +( )x n = b and the system either has a unique solution or infinitely many solutions. 60
NOTE: In this case, rank of augmented matrix = the rank of coefficient matrix 61
Theorem: A system of linear equations is consistent if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. 62
5.4. Reduced Row Echelon Form Example Solve the system (c.f. Example 1) x + 2y 5z = 1 3x 9y + 21z = 0 x + 6y 11z = 1 Augmented matrix: 1 2 5 1 3 9 21 0 1 6 11 1 Row reduce to: 1 2 5 1 0 1 2 1 0 0 1 1 63
Corresponding (equivalent) system of equations x + 2y 5z = 1 y 2z = 1 z = 1 Back substitute to get: x = 4, y = 1, z = 1. 64
Or, continue row operations: 1 2 5 1 0 1 2 1 0 0 1 1 1 0 0 4 0 1 0 1 0 0 1 1 Corresponding system of equations x = 4 y = 1 z = 1 65
Reduced Row Echelon Form 1. Rows consisting entirely of zeros are at the bottom of the matrix. 2. The first nonzero entry in a nonzero row is a 1. 3. The leading 1 in row i + 1 is to the right of the leading 1 in row i. 4. The leading 1 is the only nonzero entry in its column. 66
2. Solve the system (c.f. Example 4) 2x 1 + 5x 2 5x 3 7x 4 = 7 x 1 + 2x 2 3x 3 4x 4 = 2 3x 1 6x 2 + 11x 3 + 16x 4 = 0 Augmented matrix: 2 5 5 7 7 1 2 3 4 2 3 6 11 16 0. Row echelon form: 1 2 3 4 2 0 1 1 1 3 0 0 1 2 3. 67
Corresponding system of equations: x 1 + 2x 2 3x 3 4x 4 = 2 x 2 + x 3 + x 4 = 3 x 3 + 2x 4 = 3 Solution set: x 1 = 11 4a, x 2 = a, x 3 = 3 2a, x 4 = a, a any real number. 68
Alternative solution: Reduced row echelon form: 1 2 3 4 2 0 1 1 1 3 0 0 1 2 3 69
1 0 0 4 11 0 1 0 1 0 0 0 1 2 3 Corresponding system of equations: x 1 + 4x 4 = 11 x 2 x 4 = 0 x 3 + 2x 4 = 3 x 1 = 11 4a, x 2 = a, x 3 = 3 2a, x 4 = a, a any real number. 70
The solution set can also be written as: x 1 x 2 x 3 x 4 = a 4 1 2 1 + 11 0 3 6 71
Homogeneous Systems The system of linear equations 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. =. a m1 x 1 + a m2 x 2 + + a mn x n = b m is homogeneous if b 1 = b 2 = = b m = 0, otherwise, the system is nonhomogeneous. C.f. Linear differential equations. 72
A homogeneous system a 11 x 1 + a 12 x 2 + + a 1n x n = 0 a 21 x 1 + a 22 x 2 + + a 2n x n = 0..... a m1 x 1 + a m2 x 2 + + a mn x n = 0 ALWAYS has at least one solution, namely x 1 = x 2 = = x n = 0, called the trivial solution That is, homogeneous systems are always CONSISTENT. 73
3. Solve the homogeneous system x 2y + 2z = 0 4x 7y + 3z = 0 2x y + 2z = 0 Augmented matrix: 1 2 2 0 4 7 3 0 2 1 2 0 Row echelon form: 1 2 2 0 0 1 5 0 0 0 1 0 74
Corresponding system of equations: x 2y + 2z = 0 y 5z = 0 z = 0. This system has the unique solution x = 0, y = 0, z = 0. The trivial solution is the only solution. 75
4. Solve the homogeneous system 3x 2y + z = 0 x + 4y + 2z = 0 7x + 4z = 0 Augmented matrix: 3 2 1 0 1 4 2 0 7 0 4 0 Row echelon form: 1 4 2 0 0 1 5/14 0 0 0 0 0 76
Corresponding system of equations: x + 4y + 2z = 0 y + 5 14 z = 0 This system has infinitely many solutions: x = 4 7 a, y = 5 14 a, z = a, a any real number. 77
5. 2x 1 + 5x 2 5x 3 + 7x 4 + 5x 5 = 2 x 1 + 2x 2 3x 3 + 4x 4 + 2x 5 = 3 3x 1 6x 2 + 9x 3 10x 4 2x 5 = 0 Augmented matrix: 1 2 3 4 2 3 2 5 5 7 5 2 3 6 9 10 2 3. 78
Reduced row echelon form: 1 0 5 0 12 7 0 1 1 1 1 4 0 0 0 1 2 3 79
Corresponding system of equations: x 1 5x 3 12x 5 = 7 x 2 + x 3 + 3x 5 = 1 x 4 + 2x 5 = 3 Solution set: x 1 = 7 + 5a + 12b, x 2 = 1 a 3b x 3 = a x 4 = 3 2b, x 5 = b a, b arbitrary real nos. 80
Write as an ordered quintuple: x 1 x 2 x 3 x 4 x 5 = 7 + 5a + 12b 1 a 3b a 3 2b b = a 5 1 1 0 0 + b 12 3 0 2 1 + 7 1 0 3 0 81
What can you say about 7 1 0 3 0, and 5 1 1 0 0, 12 3 0 2 1? 82
a 5 1 1 0 0 + b 12 3 0 2 1 is the set of all solutions of the homogeneous system 2x 1 + 5x 2 5x 3 + 7x 4 + 5x 5 = 0 x 1 + 2x 2 3x 3 + 4x 4 + 2x 5 = 0 3x 1 6x 2 + 9x 3 10x 4 2x 5 = 0 83
7 1 0 3 0 is a solution of the given nonhomogeneous system 2x 1 + 5x 2 5x 3 + 7x 4 + 5x 5 = 2 x 1 + 2x 2 3x 3 + 4x 4 + 2x 5 = 3 3x 1 6x 2 + 9x 3 10x 4 2x 5 = 0 84
That is, the set of solutions of the nonhomogeneous system consists of the set of all solutions of the corresponding homogeneous system plus one solution of the nonhomogeneous system. C.f. Section 3.4. 85
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. m n is called the size of the matrix, and the numbers m and n are its dimensions. 86
A matrix in which the number of rows equals the number of columns, m = n, is called a square matrix of order n. 87
If A is an m n matrix, then a ij denotes the element in the i th row and j th column of A: A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33.... a m1 a m2 a m3 a 1n a 2n a 3n a mn. The notation A = (a ij ) also represents this display. 88
Special Cases: Vectors A 1 n matrix v = (a 1 a 2... a n ) also written as v = (a 1, a 2,..., a n ) is called an row vector. 89
An m 1 matrix v = a 1 a 2. a m is called a column vector. The entries of a row or column vector are called the components of the vector. 90
Arithmetic of Matrices Let A = (a ij ) be an m n matrix and let B = (b ij ) be a p q matrix. Equality: A = B if and only if 1. m = p and n = q; 2. a ij = b ij for all i and j. That is, A = B if and only if A and B are identical. 91
Example: a b 3 2 c 0 = 7 1 x 2 4 0 if and only if a = 7, b = 1, c = 4, x = 3. 92
Addition: Let A = (a ij ) and B = (b ij ) be m n matrices. A+B is the m n matrix C = (c ij ) where c ij = a ij + b ij for all i and j. That is, A + B = (a ij + b ij ). Addition of matrices is not defined for matrices of different sizes. 93
Examples: (a) 2 4 3 2 5 0 + 4 0 6 1 2 0 = 2 4 3 1 7 0 (b) 2 4 3 2 5 0 + 1 3 5 3 0 6 is not defined. 94
Properties: Let A, B, and C be matrices of the same size. Then: 1. A+B = B +A (Commutative) 2. (A + B) + C = A + (B + C) (Associative) A matrix with all entries equal to 0 is called a zero matrix. E.g., 0 0 0 0 0 0 and 0 0 0 0 0 0 95
The symbol 0 will be used to denote the zero matrix of arbitrary size. 3. A + 0 = 0 + A = A. The zero matrix is the additive identity. 96
The negative of a matrix A, denoted by A is the matrix whose entries are the negatives of the entries of A. 4. A + ( A) = 0. Subtraction: Let A = (a ij ) and B = (b ij ) be m n matrices. Then A B = A + ( B). 97
Example: 2 4 3 2 5 0 4 0 6 1 2 0 = 6 4 9 3 3 0. 98
Multiplication of a Matrix by a Number The product of a number k and a matrix A, denoted ka, is given by ka = (ka ij ). This product is also called multiplication by a scalar. 99
Example: 3 2 1 4 1 5 2 4 0 3 = 6 3 12 3 15 6 12 0 9. 100
Properties: Let A, B be m n matrices and let α, β be real numbers. Then 1. 1 A = A 2. 0 A = 0 3. α(a + B) = α A + α B 4. (α + β)a = α A + β A 101
Matrix Multiplication 1. The Product of a Row Vector and a Column Vector: The product of a 1 n row vector and an n 1 column vector is the number given by (a 1, a 2, a 3,..., a n ) b 1 b 2 b 3. b n = a 1 b 1 + a 2 b 2 + a 3 b 3 + + a n b n. 102
Also called scalar product (because the result is a number (scalar)), dot product, and inner product. The product of a row vector and a column vector (of the same dimension and in that order!) is a number. The product of a row vector and a column vector of different dimensions is not defined. 103
Examples (3, 2, 5) 1 4 1 = 3( 1) + ( 2)( 4) + 5(1) = 10 ( 2, 3, 1, 4) 2 4 3 5 = 2(2)+3(4)+( 1)( 3)+4( 5) = 9 104
2. Matrix Multiplication: Let A = (a ij ) be an m p matrix and let B = (b ij ) be a p n matrix. The matrix product of A and B (in that order), denoted AB, is the m n matrix C = (c ij ), where c ij is the product of the i th row of A and the j th column of B. 105
Let A and B be given matrices. The product AB, in that order, is defined if and only if the number of columns of A equals the number of rows of B. If the product AB is defined, then the size of the product is: (no. of rows of A) (no. of columns of B): A m p B = C p n m n 106
Examples: 1. A = 1 4 2 3 1 5, B = 3 0 1 2 1 2 AB: AB = 1 4 2 3 1 5 3 0 1 2 1 2 = 1 4 13 8 BA: 3 0 1 2 1 2 1 4 2 3 1 5 = 3 12 6 5 2 8 5 2 8. 107
2. A = 1 2 3 4 AB =, B = 5 1 12 9 3 26 1 1 2 3 0 5 BA does not exist. 3. A = AB = 0 0 0 0 2 1 4 2, B =, BA = 1 1 2 2 2 1 4 2 Conclusion: Matrix multiplication is not commutative; AB BA in general. 108
Properties of Matrix Multiplication: Let A, B, and C be matrices. 1. AB BA in general; NOT COMMUTATIVE. 2. (AB)C = A(BC) matrix multiplication is associative. 109
Identity Matrices: Let A be a square matrix of order n. The entries a 11, a 22, a 33,..., a nn form the main diagonal of A. For each positive integer n > 1, let I n denote the square matrix of order n whose entries on the main diagonal are all 1, and all other entries are 0. The matrix I n is called the n n identity matrix. 110
I 2 = 1 0 0 1, I 3 = 1 0 0 0 1 0 0 0 1, I 4 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 and so on. 111
3. If A is an m n matrix, then I m A = A and AI n = A. If A is a square matrix of order n, then AI n = I n A = A, 4. Inverse????? 112
Distributive Laws: 1. A(B + C) = AB + AC. This is called the left distributive law. 2. (A+B)C = AC + BC. This is called the right distributive law. 3. k(ab) = (ka)b = A(kB) 113
Quiz 4 A = 2 3 1 4 2 0 5 8 2, B = 3 1 4 2 7 6 If C = AB, then (a) c 22 = (b) c 13 = 114
Other ways to look at systems of linear equations. A system of m linear equations in n unknowns: 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 a 31 x 1 + a 32 x 3 + + a 3n x n = b 3... a m1 x 1 + a m2 x 2 + + a mn x n = b m 115
Because of the way we defined multiplication, a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31 a 32 a 33.... a 3n. a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m or in the vector-matrix form Ax = b c.f. ax = b. (1) Solution: x = A 1 b????? 116
Square matrices 1. Inverse Let A be an n n matrix. An n n matrix B with the property that AB = BA = I n is called the multiplicative inverse of A or, more simply, the inverse of A. 117
Uniqueness: If A has an inverse, then it is unique. That is, there is one and only one matrix B such that AB = BA = I. B is denoted by A 1. 118
Procedure for finding the inverse: A = 1 2 3 4 We want 1 2 3 4 x y z w = 1 0 0 1 119
Examples: 1. A = 1 2 1 0 3 4 0 1 1 2 3 4 1 0 2 1 0 1 3 2 1 2 A 1 = 2 1 3 2 1 2 120
2. B = 2 1 4 2 2 1 1 0 4 2 0 1 B does not have an inverse. 121
NOTE: Not every nonzero n n matrix A has an inverse! 122
3. C = 1 0 2 2 1 3 4 1 8 Form the augmented matrix: 1 0 2 1 0 0 2 1 3 0 1 0 4 1 8 0 0 1 123
1 0 2 1 0 0 2 1 3 0 1 0 4 1 8 0 0 1 1 0 0 11 2 2 0 1 0 4 0 1 0 0 1 6 1 1 C 1 = 11 2 2 4 0 1 6 1 1 124
Finding the inverse of A. Let A be an n n matrix. a. Form the augmented matrix (A I n ). b. Reduce (A I n ) to reduced row echelon form. If the reduced row echelon form is (I n B), then B = A 1 If the reduced row echelon form is not (I n B), then A does not have an inverse. 125
Application: Solve the system x +2z = 3 2x y +3z = 5 4x +y +8z = 2 Written in matrix form: 1 0 2 2 1 3 4 1 8 x y z = 3 5 2 Solution: Cx = b; x = C 1 b x y z = 11 2 2 4 0 1 6 1 1 3 5 2 = 29 14 15 126
2. Determinants A. Calculation 1. 2 2 a b c d = ad bc. 127
2. 3 3 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 c 3 a 1 b 3 c 2 a 2 b 1 c 3 +a 2 b 3 c 1 +a 3 b 1 c 2 a 3 b 2 c 1 a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) 128
= a 1 b 2 b 3 c 2 c 3 a 2 b 1 b 3 c 1 c 3 +a 3 b 1 b 2 c 1 c 2 This is called the expansion across the frist row. 129
a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) = b 1 (a 2 c 3 a 3 c 2 )+b 2 (a 1 c 3 a 3 c 1 ) b 3 (a 1 c 2 a 2 c 1 ) = b 1 a 2 a 3 c 2 c 3 + b 2 a 1 a 3 c 1 c 3 b 3 a 1 a 2 c 1 c 2 Expansion across the second row. 130
a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) = a 3 b 1 c 2 a 3 b 2 c 1 b 3 a 1 c 2 +b 3 a 2 c 1 +c 3 a 1 b 2 c 3 a 2 b 1 = a 3 (b 1 c 2 b 2 c 1 ) b 3 (a 1 c 2 a 2 c 1 )+c 3 (a 1 b 2 a 2 b 1 ) = a 3 b 1 b 2 c 1 c 2 b 3 a 1 a 2 c 1 c 2 + c 3 a 1 a 2 b 1 b 2 Expansion down the third column 131
and so on. You can expand across any row, or down any column. BUT: Associated with each position is an algebraic sign: + + + + + For example, across the second row: a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = = b 1 a 2 a 3 c 2 c 3 +b 2 a 1 a 3 c 1 c 3 b 3 a 1 a 2 c 1 c 2 132
3. 4 4 determinants a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 Sign chart + + + + + + + + 133
Cramer s Rule. Given a system of n linear equations in n unknowns: (a square 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 a 31 x 1 + a 32 x 2 + + a 3n x n = b 3... a n1 x 1 + a n2 x 2 + + a nn x n = b n x i = det A i, (provided det A 0) det A where A i is the matrix A with the i th b. column replaced by the vector 134
If det A 0, then the system has a unique solution. If det A = 0, then the system either has infinitely many soluitons or no solutions. 135
Examples: 1. Given the system x + 2y + 2z = 3 2x y + z = 5 4x + y 2z = 2 Does Cramer s rule apply? det A = 1 2 2 2 1 1 4 1 2 = 3 Yes! Cramer s Rule applies: 136
x + 2y + 2z = 3 2x y + z = 5 4x + y 2z = 2 Find y. y = 1 3 2 2 5 1 4 2 2 1 2 2 2 1 1 4 1 2 = 22 3 137
2. Given the system 2x + 7y + 6z = 1 5x + y 2z = 7 3x + 8y + 4z = 1 det A = 2 7 6 5 1 2 3 8 4 = 0 Cramer s Rule does not apply Does the system have infinitely many solutions or no solutions?? Compare with ax = b. 138
B. Properties of determinants: Let A be an n n matrix. 1. If A has a row or column of zeros, then det A = 0 Example: 1 0 2 2 0 3 4 0 8 Expand down second column: 0 2 3 4 8 + 0 1 2 4 8 0 1 2 3 2 = 0 139
2. If A is a diagonal matrix, A = det A = a 1 b 2 0 0 c 3 a 1 0 0 0 b 2 0 0 0 c 3 0, 0 0 0 c 3 +0 0 b 2 0 0 = a 1 b 2 c 3. In particular, det I n = 1 For example, I 3 : 1 0 0 0 1 0 0 0 1 = 1 140
3. If A is a triangular matrix, e.g., A = a 1 a 2 a 3 a 4 0 b 2 b 3 b 4 0 0 c 3 c 4 0 0 0 d 4 (upper triangular) Expand down first column!! Then det A = a 1 b 2 c 3 d 4 141
4. If B is obtained from by interchanging any two rows (columns), then det B = det A. 142
NOTE: If A has two identical rows (or columns), then det A = 0. 143
5. Multiply a row (column) of A by a nonzero number k to obtain a matrix B. Then det B = kdet A. 144
6. Multiply a row (column) of A by a number k and add it to another row (column) to obtain a matrix B. Then det B = det A. 145
7. Let A and B be n n matrices. Then det[ab] = det Adet B. 146
Example: Calculate 3 3 1 5 2 2 0 2 4 1 3 2 2 10 3 2 147
= 6 1 1 0 1 0 1 1 2 0 0 1 8 0 0 0 60 = 360 148
Equivalences: 1. The system of equations: Ax = b has a unique solution. 2. The reduced row echelon form of A is I n. 3. The rank of A is n. 4. A has an inverse. 5. det A 0. 149
A is nonsingular if det A 0; A is singular if det A = 0. 150
Linear Dependence/Independence in R n R 2 = {(a, b) : a, b R} the plane R 3 = {(a, b, c) : a, b, c R} 3- space R 4 = {(x 1, x 2, x 3, x 4 )} 4-space and so on 151
Let S = {v 1,v 2,,v k } be a set of vectors in R n. The set S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + + c k v k = 0. (c 1 v 1 + c 2 v 2 + +c k v k is a linear combination of v 1, v 2,...) 152
S is linearly independent if it is not linearly dependent. That is, S is linearly independent if c 1 v 1 + c 2 v 2 + + c k v k = 0 implies c 1 = c 2 = = c k = 0. 153
The set S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + + c k v k = 0. Another way to say this: The set S is linearly dependent if one of the vectors can be written as a linear combination of the others. 154
1. Two vectors v 1, v 2. Linearly dependent iff one vector is a multiple of the other. Examples: v 1 = (1, 2, 4), v 2 = ( 1, 1, 2) 2 linearly dependent: v 1 = 2v 2 155
v 1 = (2, 4, 5), v 2 = (0, 0, 0) linearly dependent: v 2 = 0v 1 v 1 = (5, 2, 0), v 2 = ( 3, 1, 9) linearly independent 156
2. v 1 = (1, 1), v 2 = (2, 3) v 3 = (3, 5) {v 1, v 2 } Dependent or independent?? {v 1, v 2, v 3 } Dependent or independent?? 157
Does there exist three numbers, c 1, c 2, c 3, not all zero such that c 1 v 1 + c 2 v 2 + c 3 v 3 = (0,0) That is, does the system of equations c 1 + 2c 2 + 3c 3 = 0 c 1 3c 2 5c 3 = 0 have nontrivial solutions? 158
A homogeneous system with more unknowns than equations always has infinitely many nontrivial solutions. Let v 1,v 2,,v k be a set of k vectors in R n. If k > n, then the set of vectors is (automatically) linearly dependent. 159
3. v 1 = (1, 1, 2), v 2 = (2, 3, 0), v 3 = ( 1, 2, 2), v 4 = (0, 4, 3) Dependent or independent?? a. {v 1, v 2, v 3, v 4 } b. {v 1, v 2, v 3 } c. {v 1, v 2 } 160
v 1 = (1, 1, 2), v 2 = (2, 3, 0), v 3 = ( 1, 2, 2) Does c 1 v 1 + c 2 v 2 + c 3 v 3 = 0 have non-trivial solutions? That is, does c 1 + 2c 2 c 3 = 0 c 1 3c 2 2c 3 = 0 2c 1 + 2c 3 = 0 have non-trivial solutions 161
Augmented matrix and row reduce: 1 2 1 0 1 3 2 0 2 0 2 0 162
NOTE: It s enough to row reduce: 1 2 1 1 3 2 2 0 2 163
Calculate the determinant 1 2 1 1 3 2 2 0 2 164
Or, row reduce 1 1 2 2 3 0 1 2 2 165
4. v 1 = ( a, 1, 1), v 2 = ( 1, 2a, 3), v 3 = ( 2, a, 2), v 4 = (3a, 2, a) For what values of a are the vectors linearly dependent? 166
4. v 1 = ( a, 1, 1), v 2 = ( 1, 2a, 3), v 3 = ( 2, a, 2), v 4 = (3a, 2, a) For what values of a are the vectors v 1, v 2, v 3 linearly dependent? a c 1 c 2 2c 3 = 0 c 1 + 2a c 2 + a c 3 = 0 c 1 + 3c 2 + 2c 3 = 0 167
Augmented matrix and row reduce: a 1 2 0 1 2a a 0 1 3 2 0 168
Or row reduce: a 1 2 1 2a a 1 3 2 169
Calculate the determinant a 1 1 1 2a 3 2 a 2 170
5. v 1 = (1, 1,2,1), v 2 = (3,2,0, 1) v 3 = ( 1, 4,4,3), v 4 = (2,3, 2, 2) a. {v 1, v 2, v 3, v 4 } dependent or independent? b. What is the maximum number of independent vectors? 171
Row reduce 1 1 2 1 3 2 0 1 1 4 4 3 2 3 2 2 172
Or, calculate the determinant. 1 1 2 1 3 2 0 1 1 4 4 3 2 3 2 2 173
Tests for independence/dependence Let S = v 1,v 2,,v k be a set of vectors in R n. Case 1: k > n : S is linearly dependent. 174
Case 2: k = n : Form the n n matrix A whose rows are v 1, v 2,,v n 1. Row reduce A: if the reduced matrix has n nonzero rows independent; one or more zero rows dependent. 175
Equivalently, solve the system of equations c 1 v 1 + c 2 v 2 + + c k v k = 0. Unique solution: c 1 = c 2 = = c n = 0 independent; infinitely many solutions dependent.
2. Calculate det A: det A 0 independent; det A = 0 dependent. Note: If v 1,v 2,,v n is a linearly independent set of vectors in R n, then each vector in R n has a unique representation as a linear combination of v 1,v 2,,v n.
Case 3: k < n : Form the k n matrix A whose rows are v 1, v 2,,v k 1. Row reduce A: if the reduced matrix has k nonzero rows independent; one or more zero rows dependent. 176
Equivalently, solve the system of equations c 1 v 1 + c 2 v 2 + + c k v k = 0. Unique solution: c 1 = c 2 = = c n = 0 independent; infinitely many solutions dependent.
Linear Dependence/Independence of Functions Let S = {f 1 (x), f 2 (x), f k (x)} be a set of functions defined on an interval I. S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 f 1 (x)+c 2 f 2 (x)+ +c k f k (x) 0 on I. 177
S is linearly independent if it is not linearly dependent. That is, S is linearly independent if c 1 f 1 (x)+c 2 f 2 (x)+ +c k f k (x) 0 on I implies c 1 = c 2 = = c k = 0. Examples: 178
1. f 1 (x) = 1, f 2 (x) = x, f 3 (x) = x 2 Suppose f 1, f 2, f 3 are linearly dependent. Then there exist 3 numbers c 1, c 2, c 3, NOT ALL 0 such that c 1 1 + c 2 x + c 3 x 2 0 That is, c 1 1 + c 2 x + c 3 x 2 = 0 for all x 179
2. f 1 (x) = sin x, f 2 (x) = cos x Suppose f 1, f 2 are linearly dependent. Then there exist 2 numbers c 1, c 2, NOT BOTH 0 such that c 1 cos x + c 2 sin x 0 That is, c 1 cos x + c 2 sin x = 0 for all x 180
3. f 1 (x) = sin x, f 2 (x) = cos x, f 3 (x) = cos(x + π/3) 181
Test for independence/dependence Suppose that the functions f 1 (x), f 2 (x), f k (x) are (k 1)-times differentiable on I. If the determinant W(x) = f 1 (x) f 2 (x) f k (x) f 1 (x) f 2 (x) f k (x) f (k 1) 1 (x) f (k 1) 2 (x) f (k 1) (x) k 0 for at least one x I, then the functions are linearly independent on I. NOTE: If W(x) 0 in I, then?????? The functions may be dependent, they may be independent!!!! 182
W(x) is called THEEEEEE
WRONSKIAN 183
Examples: f 1 = 2x+1, f 2 = x 3, f 3 = 3x+2 Wronskian: 2x + 1 x 3 3x + 2 2 1 3 0 0 0 = 0 Independent or dependent??? Does there exist 3 numbers, c 1, c 2, c 3, not all 0, such that c 1 f 1 + c 2 f 2 + c 3 f 3 0 184
f 1 (x) = x 3, f 2 (x) = x 3, x < 0 2x 3, x 0 Wronskian: For x < 0 x 3 x 3 3x 2 3x 2 = 0 For x 0 x 3 2x 3 3x 2 6x 2 = 0 185
Suppose there exist 2 numbers c 1, c 2, not both zero, such that c 1 f 1 (x) + c 2 f 2 (x) 0 186
Interpretations of a system of linear equations Given the system of m linear equations in n unknowns: a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3n x n = b 3... a m1 x 1 + a m2 x 2 + a m3 x 3 + + a mn x n = b m 187
The coefficient matrix A from R n to R m is a mapping a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31. a 32. a 33.. a 3n a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m 188
A is a linear transformation! A[x + y] = Ax + Ay and A[αx] = α Ax Example: A = ( 1 2 1 3 0 2 ) is a linear transformation from R 3 to R 2. 189
Write the system as a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31. a 32. a 33.. a 3n a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m or in the vector-matrix form Ax = b Given a vector b in R m, find a vector x in R n such that Ax = b. 190
Write the system equivalently as x 1 a 11 a 21. +x 2 a 12 a 22. + +x n a 1n a 2ṇ. = b 1 b 2. a m1 a m2 a mn b m This says, express b as a linear combination of the columns of A. 191
Eigenvalues/Eigenvectors Example: Set A = 1 3 1 1 1 1 3 3 1. A is a linear transformation from R 3 to R 3. 1 3 1 1 1 1 3 3 1 1 1 1 = 1 1 1 192
1 3 1 1 1 1 3 3 1 1 3 2 = 12 2 13 1 3 1 1 1 1 3 3 1 3 2 3 = 6 4 6 = 2 3 2 3 1 3 1 1 1 1 3 3 1 1 0 1 = 2 0 2 = 2 1 0 1 193
Let A be an n n matrix. A number λ is an eigenvalue of A if there is a non-zero vector v such that Av = λv 194
To find the eigenvalues of A, find the values of λ that satisfy det(a λ I) = 0. Example: Let A = 2 3 1 4 A λi = 2 λ 3 1 4 λ det (A λi) = 2 λ 3 1 4 λ = (2 λ)(4 λ) 3 = λ 2 6λ+5 = 0 Eigenvalues: λ 1 = 5, λ 2 = 1 195
Let A = 1 3 1 1 1 1 3 3 1 det (A λi) = 1 λ 3 1 1 1 λ 1 3 3 1 λ = λ 3 + λ 2 + 4λ 4 det (A λi) = 0 implies λ 3 + λ 2 + 4λ 4 = 0 or λ 3 λ 2 4λ + 4 = 0 196
Terminology: det(a λ I) is a polynomial of degree n, called the characteristic polynomial of A. The zeros of the characteristic polynomial are the eigenvalues of A The equation det(a λi) = 0 is called the characteristic equation of A. 197
A non-zero vector v that satisfies Av = λv is called an eigenvector corresponding to the eigenvalue λ.
Examples: 1. A = 2 2 2 1 Characteristic polynomial: det(a λ I) = 2 λ 2 2 1 λ = (2 λ)( 1 λ) 4 = λ 2 λ 6 Characteristic equation: λ 2 λ 6 = (λ 3)(λ + 2) = 0 Eigenvalues: λ 1 = 3, λ 2 = 2 198
Eigenvectors: 199
2. A = 4 1 4 0 Characteristic polynomial: det(a λ I) = 4 λ 1 4 λ = λ 2 4λ + 4 Characteristic equation: λ 2 4λ + 4 = (λ 2) 2 = 0 Eigenvalues: λ 1 = λ 2 = 2 200
Eigenvectors: 201
3. A = 5 3 6 1 Characteristic polynomial: det(a λ I) = 5 λ 3 6 1 λ = λ 2 4λ + 13 Characteristic equation: λ 2 4λ + 13 = 0 Eigenvalues: λ 1 = 2 + 3i, λ 2 = 2 3i 202
Eigenvectors: 203
NOTE: If a + b i is an eigenvalue of A with eigenvector α+β i, then a b i is also an eigenvalue of A and α β i is a corresponding eigenvector. 204
4. A = 4 3 5 1 2 1 1 3 2 Characteristic polynomial: det(a λ I) = 4 λ 3 5 1 2 λ 1 1 3 2 λ = λ 3 + 4λ 2 λ 6 Characteristic equation: λ 3 4λ 2 +λ+6 = (λ 3)(λ 2)(λ+1) = 0. Eigenvals: λ 1 = 3, λ 2 = 2, λ 3 = 1 205
Eigenvectors: 206
5. A = 4 1 1 2 5 2 1 1 2 Characteristic polynomial: det(a λ I) = 4 λ 1 1 2 5 λ 2 1 1 2 λ = λ 3 + 11λ 2 39λ + 45 Characteristic equation: λ 3 11λ 2 +39λ 45 = (λ 3) 2 (λ 5) = 0. Eigenvalues: λ 1 = λ 2 = 3, λ 3 = 5 207
Eigenvectors: 208
6. A = 3 1 1 7 5 1 6 6 2 Characteristic polynomial: det(a λ I) = 3 λ 1 1 7 5 λ 1 6 6 2 λ = λ 3 + 12λ + 16 Characteristic equation: λ 3 12λ 16 = (λ+2) 2 (λ 4) = 0. Eigenvalues: λ 1 = λ 2 = 2, λ 3 = 4 209
Eigenvectors: 210
Some Facts THEOREM If v 1, v 2,..., v k are eigenvectors of a matrix A corresponding to distinct eigenvalues λ 1, λ 2,..., λ k, then v 1, v 2,..., v k are linearly independent. 211
THEOREM Let A be a (real) n n matrix. If the complex number λ = a + bi is an eigenvalue of A with corresponding (complex) eigenvector u+iv, then λ = a bi, the conjugate of a + bi, is also an eigenvalue of A and u iv is a corresponding eigenvector. 212
THEOREM Let A be an n n matrix with eigenvalues λ 1, λ 2,..., λ n. Then det A = ( 1) n λ 1 λ 2 λ 3 λ n. That is, det A is the product of the eigenvalues of A. (The λ s are not necessarily distinct, the multiplicity of an eigenvalue may be greater than 1, and they are not necessarily real.) 213
Equivalences: (A an n n matrix) 1. Ax = b has a unique solution. 2. The reduced row echelon form of A is I n. 3. The rank of A is n. 4. A has an inverse. 5. det A 0. 6. 0 is not an eigenvalue of A 214