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MA2001 LINEAR ALGEBRALinear Systems & Gaussian EliminationNational University of Singapore Introduction Department of Mathematics Content. Assessment Linear Systems & Their Solutions Lines on the plane Linear Equation Solutions of a Linear Equation Linear System Consistency Examples Elementa...

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MA2001 LINEAR ALGEBRA Linear Systems & Gaussian Elimination National University of Singapore Department of Mathematics

Introduction 2 Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Linear Systems & Their Solutions 8 Lines on the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Linear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Solutions of a Linear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Elementary Row Operations 31 Augmented Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Row Equivalent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Row-Echelon Form 41 Row-Echelon Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Reduced Row-Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Solve Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Gaussian Elimination 58 Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Homogeneous Linear Systems

101

1

Homogeneous Linear Equations & Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2

Introduction

2 / 110

What will we learn in Linear Algebra I? • Why Linear Algebra? ◦ Linear: • •

Study lines, planes, and objects which are geometrically “flat”. The real world is too complicated. We may (have to) use “flat” objects to approximate.

◦ Algebra: • •

The objects are not as simple as numbers. The operations are not limited to addition, subtraction, multiplication and division. 3 / 110

What will we learn in Linear Algebra I? • Contents: ◦ Linear Equations & Gaussian Elimination. • •

Solve linear systems in systematical ways. Determine the number of solutions.

◦ Matrices. • •

Definition and computations. Determinant of square matrices.

◦ Vector Spaces. • • • •

Euclidean spaces. Subspaces. Bases and Dimensions. Change of Bases. 4 / 110

3

What will we learn in Linear Algebra I? • Contents: ◦ Vector Spaces Associated with Matrices. •

Row Spaces, Column Spaces and Null Spaces.

◦ Orthogonality. • •

Dot Product. Orthogonal and Orthonormal Bases.

◦ Diagonalization. • • •

Eigenvalues and Eigenvectors. Diagonalization and Orthogonal Diagonalization. Quadratic Forms and Conic Sections.

◦ Linear Transformation. • •

Definition, Ranges and Kernels. Geometric Linear Transformations. 5 / 110

Workload and Assessment • All lessons are conducted online via ZOOM. ◦ Lecture Group 1: •

Mondays and Wednesdays:

8:00–10:00am.

◦ Lecture Group 2: •

Tuesdays and Fridays:

2:00–4:00pm.

Recorded lectures will be uploaded to LumiNUS.

• Textbook: ◦ Linear Algebra — Concepts & Techniques on Euclidean Spaces. •

The E-version is available in NUS library.

◦ The lecture notes is prepared based on the textbook. ◦ Tutorial questions are taken from exercises of the textbook. •

Refer to course outline for details. 6 / 110

4

Workload and Assessment • Tutorials are conducted online via ZOOM Week 3 – Week 11. ◦ Tutorial questions are taken from exercises of the textbook. ◦ Some tutorial sessions are recorded and uploaded to LumiNUS. • Homework Assignments. ◦ Four homework are scanned and submitted to LumiNUS on •

14 February, 28 February, 21 March and 11 April (Mondays).

◦ Each homework consists of 5% of final marks. • Mid-Term Test. ◦ The test is scheduled on 5 March (Saturday) 8:30–10:00 am. ◦ It is proctoring by ZOOM and consists of 30% of final marks. • Final Exam. ◦ The exam is scheduled on 28 April (Thursday) 9:00–11:00 am. ◦ It is proctoring by ZOOM, and it consists of 50% of final marks. 7 / 110

Linear Systems & Their Solutions

8 / 110

Lines on the plane • Consider the xy -plane: y

(x0 , y0 )

y0 b

x0

O

x

◦ Every point on the xy-plane can be uniquely represented by a pair of real numbers (x0 , y0 ). 9 / 110

5

Lines on the plane • Consider the xy -plane: y

ax + by = c

x

O

◦ The points on a straight line are precisely all the points (x, y) on the xy-plane satisfying a linear equation •

ax + by = c ,

where a and b are not both zero. 10 / 110

Linear Equation • A linear equation in n variables (unknowns) x1 , x2 , . . . , xn is an equation in the form ◦

a1 x1 + a2 x2 + · · · + an xn = b ,

where a1 , a2 , . . . , an and b are real constants.

• Note: In a linear equation, we do not assume that a1 , a2 , . . . , an are not all zero. ◦ If a1 = · · · = an = 0 but b 6= 0, it is inconsistent. ◦ If a1 = · · · = an = b = 0, it is a zero equation. ◦ A linear equation which is not a zero equation is called a nonzero equation. For instance,

◦ 0x1 + 0x2 = 1 is inconsistent; ◦ 0x1 + 0x2 = 0 is a zero equation; ◦ 2x1 − 3x2 = 4 is a nonzero equation. 11 / 110

6

Examples • The following equations are linear equations: ◦ x + 3y = 7; ◦ x1 + 2x2 + 2x3 + x4 = x5 ; •

x1 + 2x2 + 2x3 + x4 − x5 = 0.

◦ y = x − 21 z + 4.5; •

−x + y + 12 z = 4.5.

• The following equations are NOT linear equations: ◦ xy = 2; ◦ sin θ + cos φ = 0.2; ◦ x12 + x22 + · · · + xn2 = 1; ◦ x = ey . 12 / 110

Examples • In the xyz -space, the linear equation ◦

ax + by + cz = d ,

where a, b, c are not all zero, represents a plane. z

O

y

x

For instance, x + y + z = 1 represents a plane in the xyz -space. 13 / 110

7

Solutions of a Linear Equation • Let a1 x1 + a2 x2 + · · · + an xn = b be a linear equation in n variables x1 , x2 , . . . , xn . ◦ For real numbers s1 , s2 , . . . , sn , if •

a1 s1 + a2 s2 + · · · + an sn = b,

then x1 = s1 , x2 = s2 , . . . , xn = sn is a solution to the given linear equation.

◦ The set of all solutions is called the solution set. •

•

The solution set of ax + by = c (in x, y ), where a, b are not all zero, represents a straight line in xy -plane. The solution set of ax + by + cz = d (in x, y, z ), where a, b, c not all zero, represents a plane in xyz -space.

◦ An expression that gives the entire solution set is a general solution. 14 / 110

Examples • Linear equation 4x − 2y = 1 in variables x and y . ◦ x can take any arbitrary value, say t. •

•

x = t ⇒ y = 2t − 21 . ( x = t, t is a parameter. General solution: y = 2t − 12 ,

◦ y can take any arbitrary value, say s. •

•

y = s ⇒ x = 21 s + 14 . ( x = 21 s + 41 , s is a parameter. General solution: y = s,

• Different representations of the same solution set.

( x = 1, ◦ y = 1.5,

( x = 1.5, y = 2.5,

( x = −1, y = −2.5,

.... 15 / 110

8

Examples • x1 − 4x2 + 7x3 = 5 in three variables x1 , x2 , x3 . ◦ x2 and x3 can be chosen arbitrarily, say s and t. •

•

x2 = s and x3 = t ⇒ x1 = 5 + 4s − 7t.    x1 = 5 + 4s − 7t,  

x2 = s, x3 = t,

s, t are arbitrary parameters.

◦ x1 and x2 can be chosen arbitrarily, say s and t. •

•

x1 = s and x2 = t ⇒ x3 = 75 − 71 s + 47 t.    x1 = s, s, t are arbitrary parameters. x2 = t,   5 − 1s + 4 t, x3 = 7 7 7

16 / 110

Examples • In xy -plane, x + y = 1 has a general solution ◦ (x, y) = (1 − s, s), s is an arbitrary parameter. These points form a line in xy -plane: y 1 (1 − s, s) b

x+y =1

O

1

x

17 / 110

9

Examples • In xyz -space, x + y = 1 has a general solution ◦ (x, y, z) = (1 − s, s, t), s, t are arbitrary parameters. These points form a plane in xyz -space: z

(1 − s, s, t) b

O

b

y

(1 − s, s, 0)

x

The projection of “the plane x + y = 1 in xyz -space” on the xy -plane is “the line x + y = 1 in

xy-plane”. 18 / 110

Examples • The zero equation in n variables x1 , x2 , . . . , xn is ◦ 0x1 + 0x2 + · · · + 0xn = 0 (or simply 0 = 0). The equation is satisfied by any values of x1 , x2 , . . . , xn .

◦ The general solution is given by •

(x1 , x2 , . . . , xn ) = (t1 , t2 , . . . , tn ),

where t1 , t2 , . . . , tn are arbitrary parameters.

• Let b 6= 0. An inconstant equation in n variables x1 , x2 , . . . , xn ◦ 0x1 + 0x2 + · · · + 0xn = b (or simply 0 = b). It is NOT satisfied by any values of x1 , x2 , . . . , xn .

◦ An inconstant equation has NO solution. 19 / 110

10

Linear System • A linear system (system of linear equations) of m linear equations in n variables x1 , x2 , . . . , xn is

  a11 x1    a x 21 1 ◦    a x m1 1

+ a12x2 + · · · + a1n xn = b1 , + a22x2 + · · · + a2n xn = b2 , .. .

...

+ am2 x2 + · · · + amn xn = bm ,

where aij and bi are real constants.

◦ aij is the coefficient of xj in the ith equation, ◦ bi is the constant term of the ith equation. • If all aij and bi are zero, ◦ the linear system is called a zero system. If some aij or bi is nonzero,

◦ the linear system is called a nonzero system. 20 / 110

Linear System • A linear system (system of linear equations) of m linear equations in n variables x1 , x2 , . . . , xn is

  a11 x1    a21 x1 ◦    a x m1 1

+ a12x2 + · · · + a1n xn = b1 , + a22x2 + · · · + a2n xn = b2 , .. .

...

+ am2 x2 + · · · + amn xn = bm ,

where aij and bi are real constants.

• If x1 = s1 , x2 = s2 , . . . , xn = sn is a solution to every equation of the linear system, then it is called a solution to the system.

◦ The solution set is the set of all solutions to the linear system. ◦ A general solution is an expression which generates the solution set of the linear system. 21 / 110

11

Example • Linear system



4x1 − x2 + 3x3 = −1, 3x1 + x2 + 9x3 = −4.

◦ x1 = 1, x2 = 2, x3 = −1 is a solution to both equations, then it is a solution to the system. ◦ x1 = 1, x2 = 8, x3 = 1 is a solution to the first equation, but not a solution to the second equation; so it is not a solution to the system. Problem: How to find a general solution to the system?

  x1 = 1 + 12t, x2 = 2 + 27t, where t is an arbitrary parameter. ◦  x = −1 − 7t, 3

22 / 110

Consistency • Remark. In a linear system, even if every equation has a solution, there may not be a solution to the system.

◦

 • •

x + y = 4, 2x + 2y = 6. 2x + 2y = 6 ⇒ x + y = 3. x + y = 4 & x + y = 3 ⇒ 4 = 3, impossible!

• Definition. A linear system is called ◦ consistent if it has at least one solution; ◦ inconsistent if it has no solution. • Remark. A linear system has either ◦ no solution, or ◦ exactly one solution, or ◦ infinitely many solutions. (To be proved in Chapter 2) 23 / 110

12

Examples • Linear system in variables x, y of two equations: ◦



a1 x + b1 y = c1 , (L1 ) a 2 x + b2 y = c 2 . ( L 2 )

Assume a1 , b1 are not both zero, a2 , b2 are not both zero.

◦ In xy -plane, each equation represents a straight line. y L1

L2

x

O

◦ The system has no solution

⇔ L1 and L2 are parallel but distinct. 24 / 110

Examples • Linear system in variables x, y of two equations: ◦



a1 x + b1 y = c1 , (L1 ) a 2 x + b2 y = c 2 . ( L 2 )

Assume a1 , b1 are not both zero, a2 , b2 are not both zero.

◦ In xy -plane, each equation represents a straight line. y

L1

L2 x

O

◦ The system has exactly one solution

⇔ L1 and L2 are not parallel. 25 / 110

13

Examples • Linear system in variables x, y of two equations: ◦



a1 x + b1 y = c1 , (L1 ) a 2 x + b2 y = c 2 . ( L 2 )

Assume a1 , b1 are not both zero, a2 , b2 are not both zero.

◦ In xy -plane, each equation represents a straight line. y

L1

L2

x

O

◦ The system has infinitely many solutions

⇔ L1 and L2 are the same line. 26 / 110

Examples • Linear system in variables x, y, z of two equations: ◦



a1 x + b1 y + c1 z = d1 , (P1 ) a2 x + b2 y + c2 z = d2 . (P2 )

Assume a1 , b1 , c1 not all zero, a2 , b2 , c2 not all zero.

◦ Each equation represents a plane in xyz -space.

P2

P1

◦ The system has no solution

⇔ P1 and P2 are parallel but distinct. 27 / 110

14

Examples • Linear system in variables x, y, z of two equations: ◦



a1 x + b1 y + c1 z = d1 , (P1 ) a2 x + b2 y + c2 z = d2 . (P2 )

Assume a1 , b1 , c1 not all zero, a2 , b2 , c2 not all zero.

◦ Each equation represents a plane in xyz -space.

P1

P2

◦ The system has infinitely many solutions if P1 and P2 are the same plane. 28 / 110

Examples • Linear system in variables x, y, z of two equations: ◦



a1 x + b1 y + c1 z = d1 , (P1 ) a2 x + b2 y + c2 z = d2 . (P2 )

Assume a1 , b1 , c1 not all zero, a2 , b2 , c2 not all zero.

◦ Each equation represents a plane in xyz -space. P2

P1

◦ The system has infinitely many solutions if P1 and P2 intersect at a straight line. 29 / 110

15

Examples • Linear system in variables x, y, z of two equations: ◦



a1 x + b1 y + c1 z = d1 , (P1 ) a2 x + b2 y + c2 z = d2 . (P2 )

Assume a1 , b1 , c1 are not all zero, a2 , b2 , c2 are not all zero.

◦ Each equation represents a plane in xyz -space.

P1 and P2 represent the same plane

1.

⇔ a1 : a2 = b1 : b2 = c1 : c2 = d1 : d2 . P1 and P2 are parallel planes

2.

⇔ a 1 : a 2 = b1 : b2 = c 1 : c 2 . P1 and P2 intersect at a line

3.

⇔ a1 : a2 , b1 : b2 , c1 : c2 are not all the same. 30 / 110

Elementary Row Operations

31 / 110

Augmented Matrix • A linear system in variables x1 , x2 , . . . , xn :

  a11 x1    a x 21 1 ◦    a x m1 1

+ a12x2 + · · · + a1n xn = b1 , + a22x2 + · · · + a2n xn = b2 , .. .

...

+ am2 x2 + · · · + amn xn = bm ,

◦ The rectangular array of constants

•



a11 a12 · · · a1n b1  a21 a22 · · · a2n b2   .. .. ... ...  . . am1 am2 · · · amn bm

    

is called the augmented matrix of the linear system.

• A linear system in y1 , y2 , . . . , yn with the same coefficients & constant terms has the same augmented matrix. 32 / 110

16

Example   x1 + x2 + 2x3 = 9, 2x + 4x2 − 3x3 = 1, • Linear system  1 3x1 + 6x2 − 5x3 = 0.   1 1 2 9 ◦ Augmented matrix:  2 4 −3 1  3 6 −5 0 This is also the augmented matrix for:

•

•

  y1 + y2 + 2y3 = 9, 2y + 4y2 − 3y3 = 1,  1 3y1 + 6y2 − 5y3 = 0.   ♠ + ♥ + 2♣ = 9, 2♠ + 4♥ − 3♣ = 1,  3♠ + 6♥ − 5♣ = 0.

33 / 110

Elementary Row Operations • To solve a linear system, we perform operations: ◦ Multiply an equation by a nonzero constant. ◦ Interchange two equations. ◦ Add a constant multiple of an equation to another. • •

E1 7→ E1 + cE2 = E3 . E3 7→ E3 + (−c)E2 = E1 .

• In terms of augmented matrix, they correspond to operations on the rows of the augmented matrix: ◦ Multiply a row by a nonzero constant. ◦ Interchange two rows. ◦ Add a constant multiple of a row to another row. • •

R1 → 7 R1 + cR2 = R3 . R3 → 7 R3 + (−c)R2 = R1 . 34 / 110

17

Elementary Row Operations • The operations on rows of an augmented matrix: ◦ Multiply a row by a nonzero constant; ◦ Interchange two rows; ◦ Add a constant multiple of a row to another row; are called the elementary row operations.

• Remark. Interchanging two rows can be obtained by using the other two operations.     R1 add 2nd row to 1st row R1 + R2 −−−−−−−−−−−→ R2 R2   add (−1) times 1st row to 2nd row R1 + R2 −−−−−−−−−−−−−−−−−→ −R1   multiply 2nd row by (−1) R1 + R2 −−−−−−−−−−−−→ R1   add (−1) times 2nd row to 1st row R2 −−−−−−−−−−−−−−−−−→ R1

35 / 110

Example • Compare operations of equations in a linear system and corresponding row operations of augmented matrix.

  x + y + 3z = 0 (1) 2x − 2y + 2z = 4 (2) ◦  3x + 9y = 3 (3) •

•