Linear Algebra (BAS012) Lecture Notes PDF

LINEAR ALGEBRA (BAS012) SPHINX UNIVERSITY Dr.Amira A.Allam [email protected]  Practical Information Lectures 12 lectures (2 hours) Tutorials 9 sheets (2 hours) Readings Stewart, J. (2015) Calculus, Brooks/Cole Publishing Company,....

LINEAR ALGEBRA (BAS012) SPHINX UNIVERSITY Dr.Amira A.Allam [email protected]  Practical Information Lectures 12 lectures (2 hours) Tutorials 9 sheets (2 hours) Readings Stewart, J. (2015) Calculus, Brooks/Cole Publishing Company,.8th edition. Briggs, Cochran and Gillett (2015), Calculus, Pearson; 2nd edition. Syllabus Complex Numbers Synthetic division and the roots (Numerical root finder) Binomial series mathematical induction vector spaces Matrices (Gauss elimination method - inverse of a nonsingular - eigenvalues and eigenvectors – transformation matrices)  Assessment System 40% Final exam 30% Mid-term exam Semester work 20% Quiz 10% Tutorial  Important Remarks  Lectures notes are available, while white board is an essential element of this course.  Take notes in the lectures (just copy the white board).  Attempt all homework problems.  Review the material before each exam  textbook  Lectures notes  Model answer of the homework problems  Lecture 1  Complex numbers  operations on Complex numbers  conjugate of Complex numbers Complex numbers Definition: A complex number 𝑧 is an ordered pair of real numbers 𝑎, 𝑏 : a is called Real part of 𝑧, denoted by 𝑅𝑒 𝑧 and 𝑏 is called imaginary part of 𝑧, denoted by 𝐼𝑚 𝑧. If 𝑅𝑒 𝑧 = 0, then z is called purely imaginary If 𝐼𝑚 𝑧 = 0, then z is called real. 𝐶 is the set of all complex numbers. Complex numbers operations of addition and multiplication over the set C of all complex numbers (𝑎, 𝑏) = (𝑐, 𝑑) if and only if 𝑎 = 𝑐 𝑎𝑛𝑑 𝑏 = 𝑑, (𝑎, 𝑏) + (𝑐, 𝑑) = (𝑎 + 𝑐, 𝑏 + 𝑑), (𝑎, 𝑏). (𝑐, 𝑑) = (𝑎𝑐 − 𝑏𝑑, 𝑎𝑑 + 𝑏𝑐) operations on Complex numbers The set 𝐶 of all complex numbers under the operations of addition and multiplication as defined above satisfies following properties: for 𝑧1 , 𝑧2 , 𝑧3 ∈ 𝐶, 1) 𝑧1 + 𝑧2 = 𝑧2 + 𝑧1 (commutatively). 2) (𝑧1 + 𝑧2 ) + 𝑧3 = 𝑧1 + (𝑧2 + 𝑧3 ) (associativity), 3) 𝑧1 + (0,0) = 𝑧1 (existence of identity), 4) for 𝑧 = (𝑎, 𝑏) ∈ 𝐶, there exists – 𝑧 = (−𝑎, −𝑏) ∈ 𝐶 such that (−𝑧) + 𝑧 = 𝑧 + (−𝑧) = (0,0) (existence of inverse), operations on Complex numbers for 𝑧1 , 𝑧2 , 𝑧3 ∈ 𝐶, 1) 𝑧1. 𝑧2 = 𝑧2. 𝑧1 (commutatively). 2) 𝑧1. 𝑧2. 𝑧3 = 𝑧1. (𝑧2. 𝑧3 ) (associativity), 3) 𝑧1. (1,0) = 𝑧1 (existence of identity), 1 4) for 𝑧 = (𝑎, 𝑏) ∈ 𝐶, and 𝑧 ≠ (0,0) there exists ∈ 𝐶 such that 𝑧 1 1. 𝑧 = 𝑧. = 1 (existence of inverse), 𝑧 𝑧 operations on Complex numbers Denoting the complex number (0,1) by 𝑖 and identifying a real complex number (𝑎, 0) with the real number 𝑎, we see 𝑧 = (𝑎, 𝑏) = (𝑎, 0) + (0, 𝑏) = (𝑎, 0) + (0,1)(𝑏, 0) can be written 𝑎𝑠 𝑧 = 𝑎 + 𝑖𝑏. For two real numbers 𝑎, 𝑏 , 𝑎2 + 𝑏 2 = 0 implies 𝑎 = 0 = 𝑏, but in complex numbers not the same, for example 12 + 𝑖 2 = 0 but 1 ≡ (1,0) ≠ (0,0) ≡ 0 𝑎𝑛𝑑 𝑖 = (0,1) ≠ 0,0. 𝑖 2 = (0,1)(0,1) = (−1,0) ≡ −1. Conjugate of Complex numbers Let 𝑧 = (𝑥, 𝑦) ≡ 𝑥 + 𝑖𝑦. The conjugate of 𝑧, denoted by 𝑧 , is (𝑥, −𝑦) ≡ 𝑥 − 𝑖𝑦. Conjugate of Complex numbers Geometrically, the point (representing) 𝑧 is the reflection of the point (representing) 𝑧 in the real axis. The conjugation operation satisfies the following properties: 𝑧=𝑧 𝑧1 + 𝑧2 = 𝑧1 + 𝑧2 𝑧 𝑧1 ( 1) = 𝑧2 𝑧2 𝑧 + 𝑧 = 2 𝑅𝑒 𝑧, 𝑧 − 𝑧 = 2𝑖 𝐼𝑚(𝑧) Exercises Write down the imaginary part of the complex number 3 + 4𝑖 − 6(2 + 8𝑖). Answer: Exercises Write down the imaginary part of the complex number 3 + 4𝑖 − 6(2 + 8𝑖). Answer: 𝐼𝑚*3 + 4𝑖 − 6(2 + 8𝑖)+ = 𝐼𝑚*3 − 12 + (4 − 48)𝑖+ = 4 − 48 = −44 Exercises Solve the equation 𝑥 2 + 4𝑥 + 9 = 0 simplifying your answers as far as possible. Answer: Exercises Solve the equation 𝑥 2 + 4𝑥 + 9 = 0 simplifying your answers as far as possible. Answer: (𝑥 + 2)2 −4 + 9 = 0 ⇒ (𝑥 + 2)2 = −5 ⇒ 𝑥 = −2 ± 𝑖 5 Exercises Solve the equation 16𝑥 4 = 625. Answer: 625 625 𝑥4 = 4 ⇒𝑥 − =0 16 16 2 25 2 25 ⇒ (𝑥 − )(𝑥 + ) = 0 4 4 5 5 5 5 ⇒ (𝑥 − )(𝑥 + )(𝑥 + 𝑖)(𝑥 − 𝑖) = 0 2 2 2 2 5 5 ⇒ 𝑥 = ± 𝑜𝑟 ± 𝑖 2 2