Chapter 5. Linear Algebra Sections 5.1 5.3 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 equations 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 equations in two unknowns x, y: ax + by = α. A solution of the equation is an ordered pair of numbers (x, y). Assuming 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). 4
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), that is, the system has exactly one solution 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) that is, the system has no solution 3 2 1-3 -2-1 1 2 3 7
(c) coincide (same slope, same y- intercept), that is, the system has infinitely many solutions. 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). 13
Assuming 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 14
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), they coincide (infinitely many solutions, a whole plane of solutions), 15
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 + 4z = 12 2x y + 5z = 18 x + 3y 3z = 8 21
Example 2 con t 22
Equivalent system x 2y + 4z = 12 y z = 2 z = 3 Solution set: x = 2, y = 1, z = 3 23
Example 3: Solve the system 3x 4y z = 3 2x 3y + z = 1 x 2y + 3z = 2 24
Example 3 con t 25
Equivalent system x 2y + 3z = 2 y 5z = 3 0z = 1 The system has no solution. 26
The Elementary Operations The operations that produce equivalent systems are called elementary operations. 1. Multiply/divide an equation by a nonzero number. 2. Interchange two equations. 3. Multiply an equation by a number and add it to another equation. 27
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 + 3x 4 = 5 28
Example 4 con t 29
Equivalent system x 1 2x 2 + x 3 x 4 = 2 x 2 + x 3 + 2x 4 = 3 x 3 + 4x 4 = 1 30
Solution set: x 1 = 5 + 9a, x 2 = 2 + 2a, x 3 = 1 4a, x 4 = a, a any real number. 31
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 32
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 33
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. 34
Examples 5. Solve the system (same as Ex. 2) x 2y + 4z = 12 2x y + 5z = 18 x + 3y 3z = 8 Augmented matrix: 1 2 4 12 2 1 5 18 1 3 3 8 35
Row reduce 1 2 4 12 2 1 5 18 1 3 3 8 36
1 2 4 12 0 1 1 2 0 0 1 3 Corresponding (equivalent) system of equations: x 2y + 4z = 12 y z = 2 z = 3 Solution set: x = 2, y = 1, z = 3 37
6. Solve the system (same as Ex. 3) 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. 38
Row reduce 3 4 1 3 2 3 1 1 1 2 3 2 39
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. 40
7. 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 41
Row reduce 1 1 3 1 2 1 4 0 3 2 1 7 42
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 43
This system has infinitely many solutions given by: x = 1 + a, y = 2 + 2a, z = a, a any real number. 44
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. 45
1 2 4 12 0 1 1 2 0 0 1 3 (Example 5) 1 2 3 2 0 1 5 3 0 0 0 1 (Example 6) 1 1 3 1 0 1 2 2 0 0 0 0 (Example 7) 46
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. 47
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. 48
4. Back substitute to find the solution set. This method is called Gaussian elimination with back substitution. 49
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. 50
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. 51
8 (see 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 + 3x 4 = 5 Augmented matrix: 1 2 1 1 2 2 5 1 4 1 3 7 4 3 5 52
Transform to row echelon form: 53
9. 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. 54
Transform to row echelon form: 2 5 5 7 8 1 2 3 4 2 3 6 11 16 0. 55
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 56
Solution set: x 1 = 9 4a, x 2 = 1 + a, x 3 = 3 2a, x 4 = a, a any real number. 57
9. 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 58
Transform to row echelon form: 1 3 2 1 2 2 3 9 7 1 3 7 2 6 7 4 5 7 59
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 60
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 61
10. 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? 62
1 1 1 1 2 3 k 3 1 k 3 2 63
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. 64
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. 65
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. 66
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. 67
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. 68
NOTE: In this case, rank of augmented matrix = the rank of coefficient matrix 69
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. 70
5.4. Reduced Row Echelon Form Examples 1. Solve the system (See Example 5, pg. 35) x 2y + 4z = 12 2x y + 5z = 18 x + 3y 3z = 8 Augmented matrix: 1 2 4 12 2 1 5 18 1 3 3 8 71
Row reduce to: 1 2 4 12 0 1 1 2 0 0 1 3 Corresponding (equivalent) system of equations x 2y + 4z = 12 y z = 2 z = 3 Back substitute to get: x = 2, y = 1, z = 3.
Or, continue row operations: 1 2 4 12 0 1 1 2 0 0 1 3 Corresponding system of equations x = 2 y = 1 z = 3 72
2. Solve the system (c.f. Example 8, pg. 52) 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. Row echelon form: 1 2 3 4 2 0 1 1 1 4 0 0 1 2 3. 73
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 Solution set: x 1 = 9 4a, x 2 = 1 + a, x 3 = 3 2a, x 4 = a, a any real number. 74
Alternative solution: Continue the row operations 1 2 3 4 2 0 1 1 1 4 0 0 1 2 3 75
1 0 0 4 9 0 1 0 1 1 0 0 1 2 3 Corresponding system of equations: x 1 + 4x 4 = 9 x 2 x 4 = 1 x 3 + 2x 4 = 3 x 1 = 9 4a, x 2 = 1+a, x 3 = 3 2a, x 4 = a, a any real number. 76
3. Solve the system of equations 2x 1 +3x 2 5x 3 2x 4 = 2 2x 1 4x 2 +4x 3 3x 4 = 6 x 1 +2x 2 2x 3 +3x 4 = 0 2 3 5 2 2 2 4 4 3 6 1 2 2 3 0 1 2 2 3 0 2 4 4 3 6 2 3 5 2 2 1 2 2 3 0 0 0 0 3 6 0 1 1 8 2 1 2 2 3 0 0 1 1 8 2 0 0 0 1 2 row echelon form 77
1 2 2 3 0 0 1 1 8 2 0 0 0 1 2 1 2 2 0 6 0 1 1 0 18 0 0 0 1 2 1 0 4 0 30 0 1 1 0 18 0 0 0 1 2 78
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. 79
Examples Example 1 1 2 4 12 0 1 1 2 0 0 1 3 1 0 0 2 0 1 0 1 0 0 1 3 Example 2 1 2 3 4 2 0 1 1 1 4 0 0 1 2 3 1 0 0 4 9 0 1 0 1 1 0 0 1 2 3 Example 3 1 2 2 3 0 0 1 1 8 2 0 0 0 1 2 1 0 4 0 30 0 1 1 0 18 0 0 0 1 2 80
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. 81
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. 82
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 83
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. 84
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 85
Corresponding system of equations: x + 4y + 2z = 0 y + 5 14 z = 0 This system has infinitely many solutions: x = 2 7 a, y = 5 14 a, z = a, a any real number. 86