F07 - Matrices and Linear Spaces - Lecture Notes (Student s Version)

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 1 of 99 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Matrices and Linear Spaces (Lecture Notes) Key Questions to Answer: 1. How do we use matrices to represent a set off linear equations? 2. What are the common operations on matrices? 3. How do we find the determinant of a 2 2 or 3 3 matrix? 4. How do we find the inverse of a non-singular 2 2 or 3 3 matrix? 5. How do we use matrices to solve a set of linear equations? What is the geometrical interpretation of the solution? 6. What is a linear space? What is a subspace? 7. What are the axioms for a linear space? 8. What is a span? What is linear independence? 9. How do we find the basis and dimension of a linear space? 10. How do we find the column space, row space, range space and null space of a matrix? 11. What is the rank of a square matrix? What is the relation between the rank, dimension of null space and the order of the matrix? 12. What are linear transformations? 13. What are the eigenvalues and eigenvectors of a 2 2 or 3 3 matrix? 14. How do we diagonalize a square matrix? What are the applications of diagonalization? §1 System of Linear Equations Page 2 §2 Matrices and Matrix Operations Page 13 §3 Inverse Matrix and Its Applications Page 23 §4 Determinants Page 32 §5 Real Vector Spaces Page 42 §6 Span, Linear Independence, Basis and Dimension Page 51 §7 Row Space, Column Space and Null Space Page 65 §8 Linear Transformations Page 74 §9 Eigenvalues and Eigenvectors Page 82 Summary Pages Page 94 Appendices Page 97 I: Calculators II: Some Mathematical Terminologies III: Some Applications www.KiasuExamPaper.com 428

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 2 of 99 §1 System of Linear Equations Systems of linear equations arise in a wide variety of applications, such as polynomial curve fitting, network analysis and optimisation. You may refer to Appendix III for more details. 1.1 Linear Systems Definition A system of m linear equations in n unknown x1, x2, x3, …, x n is a set of m linear equations each in n unknowns: 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 ... ... ... , n n n n m m mn n m a x a x a x b a x a x a x b a x a x a x b              (*) where ija and ib , 1 i m  , 1 j n  are constants. A sequence of numbers 1 2, , ..., ns s s (or 1 1 2 2 , , ..., n nx s x s x s   ) is called a solution of the system (*) if every equation in the system is satisfied when we substitute 1 1 2 2 , , ..., n nx s x s x s   . Example 1.1.1 Verify that 1x , 2y  and 2z   is a solution of the linear system 5 3 7. x y z x z      Determine whether 2x , 3y and 0z is also a solution of the system. Suggest another solution of the system. Solution: Substitute 1x , 2y  and 2z   into both equations, since  1 2 2 5    and  1 3 2 7   , it is a solution of the linear system. Since  2 3 0 2 7   , 2x , 3y and 0z is not a solution of the system. Another solution can be 4x , 0y and 1z   (not unique) www.KiasuExamPaper.com 429

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 3 of 99 Example 1.1.2 Suggest the number of solution(s) of each of the following linear systems: (a) 2x y  (b) 1x y  (c) 2 1x y  4x y  2 2 6x y   2 4 2x y  Solution: (a) Exactly one solution (b) No solution (c) Infinitely many solutions Theorem 1.1.1 Every system of linear equations has either no solution, exactly one solution or infinitely many solutions. (There are no other possibilities)  The theorem is not true if the equations are not all linear. Can you give an example?  For a system of linear equations in 2 unknowns, what is the geometrical interpretation of the theorem?  For a system of linear equations in 3 unknowns, what is the geometrical interpretation of the theorem? Definition If a system of equations has no solution, they we say that it is inconsistent; if the system has at least one solution, they we say that it is consistent. In Example 1.1.2, (a) and (c) are consistent, but (b) is consistent. Example 1.1.3 Solve the following linear system by elimination 3 2 1 4 6 x y x y     Solution: 4 6 3 2 1 x y x y     (1) 4 6 14 17 x y y     (2) 4 6 17 14 x y y    (3) By backward substitution, we obtain the solution of the linear system: 8 7x and 17 14y  . www.KiasuExamPaper.com 430

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 4 of 99 Example 1.1.4 Solve the following linear system by elimination 3 2 5 2 2 5 0 x y x y z x y z          Solution: 3 2 2 5 4 2 5 0 x y y z x y z         (1) 3 2 2 5 4 4 x y y z y z        (2) 3 2 4 2 5 4 x y y z y z        (3) 3 2 4 7 4 x y y z z      (4) 3 2 4 4 7 x y y z z      (5) By backward substitution, we obtain the solution of the linear system: 58 7x , 24 7y   and 4 7z .  In the processes of solving Example 1.1.3 and Example 1.1.4, what types of operations have we performed in each step? Note that the method of elimination is to simplify a system of linear equations to another system of linear equations that has exactly the same set of solution(s), but is easier to solve. In the method of elimination, we perform the following three types of operations: 1. Multiply an equation through by a nonzero constant. 2. Interchange two equations. 3. Add a multiple of one equation to another. www.KiasuExamPaper.com 431

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 5 of 99 1.2 Gaussian and Gauss-Jordan Elimination Definition Given a linear system (*) above, the rectangle array of numbers 11 12 1 1 21 22 2 2 1 2 ... n n m m mn m a a a b a a a b a a a b                   is called the augmented matrix of the linear system (*). Example 1.2.1 Write down the augmented matrix of each of the following linear systems: (a) 2 5 2 3 4 7 3 2 3 x z x y z x y         (b) 1 2 3 x y z    Solution: (a) 2 0 1 5 2 3 4 7 3 2 0 3        (b) 1 0 0 1 0 1 0 2 0 0 1 3         Definition Corresponding to the three types of operations in the method of elimination, the following operations on the rows of the augmented matrix are called elementary row operations: 1. Multiply a row through by a nonzero constant. 2. Interchange two rows. 3. Add a multiple of one row to another row. Example 1.2.2 Solve the linear system in Example 1.1.4 by performing elementary row operations: 3 2 5 2 2 5 0 x y x y z x y z          www.KiasuExamPaper.com 432

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Student’s Version) Page 6 of 99 Solution: The augmented matrix of the linear system is 1 3 0 2 1 1 5 2 2 5 1 0        .  3 2 12 1 2 3 3 3 2 2 7 4 7 1 3 0 2 1 3 0 2 1 3 0 2 1 3 0 2 1 1 5 2 0 2 5 4 0 2 5 4 0 1 1 4 2 5 1 0 2 5 1 0 0 1 1 4 0 2 5 4 1 3 0 2 1 3 0 2 0 1 1 4 0 1 1 4 0 0 7 4 0 0 1 R RR R R R R R R                                                                             By backward substitution, we obtain the solution of the linear system: 58 7x , 24 7y   and 4 7z . C