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 a