lecture02 - Lecture 2 1.1 Linear Systems and their...

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Chapter 1 Linear Systems & Gaussian Elimination 1 Lecture 2 1.1 Linear Systems and their solutions 1.2 Elementary Row Operations
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Chapter 1 Linear Systems & Gaussian Elimination 2 Announcement ± Tutorial and lab balloting begins tomorrow ± Last chance to sign up virtual tutorial class ± Hardcopies of introductory handouts ± Watch introductory lecture on webcast ± Practice Session starts next week: SL1 lecture group : Tuesday SL2 lecture group : Thursday
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Chapter 1 Linear Systems & Gaussian Elimination 3 Section 1.1 Linear Systems and their solutions Objective •Whatisa linear equation and a linear system ? general solution of a LE/LS? • What is the geometrical interpretation? • How to find a general solution of a LE? Other terminologies consistent / inconsistent LS
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Chapter 1 Linear Systems & Gaussian Elimination 4 Discussion 1.1.1 A line in the xy -plane e.g. x + y = 1 x = 2 y = -3 is represented algebraically by a linear equation in the variables x and/or y General form ax + by = c a and b are not both zero a, b, c represent some real numbers constants y is hidden x is hidden What is a linear equation?
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Chapter 1 Linear Systems & Gaussian Elimination 5 Definition 1.1.2 variables: x 1 , x 2 , …, x n constants: a 1 , a 2 , …, a n and b A linear equation in 3 variables ax + by + cz = d A linear equation in 4 variables ax + by + cz + dw = e A linear equation in n variables a 1 x 1 + a 2 x 2 + ··· + a n x n = b also called the unknowns geometrical meaning: plane geometrical meaning: none coefficients constant term What is a linear equation?
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Chapter 1 Linear Systems & Gaussian Elimination 6 Example 1.1.3.1 a) x + 3 y = 7 b) x 1 + 2 x 2 + 2 x 3 + x 4 = x 5 c) y = x –0.5 z + 4.5 d) x 1 + x 2 + ··· + x n = 1 The following are (specific) linear equations: What is a linear equation? a 1 x 1 + a 2 x 2 + ··· + a n x n = b
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Chapter 1 Linear Systems & Gaussian Elimination 7 Example 1.1.3.2 a) xy = 2 b) sin( q ) + cos ( f ) = 0.2 c) x 1 2 + x 2 2 + ··· + x n 2 = 1 d) x = e y The following are not linear equations: cross term square terms function of y not linear in q and f linear in sin( q ) and cos( f ) What is a linear equation? a 1 x 1 + a 2 x 2 + ··· + a n x n = b
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Chapter 1 Linear Systems & Gaussian Elimination 8 Example 1.1.3.3 ax + by + cz = d not all a , b , c are zero represents a plane in the three dimensional space 0 x + 0 y + cz = d horizontal plane ( ^ z-axis) 0 x + by + 0 z = d vertical plane ( ^ y-axis) ax + 0 y + 0 z = d vertical plane ( ^ x-axis) If d = 0, the plane passes through origin If a, b, c all non-zero, the plane is “slanting” x y z What is a linear equation? What if there is only one zero coefficient?
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Chapter 1 Linear Systems & Gaussian Elimination 9 Definition 1.1.4 a 1 x 1 + a 2 x 2 + ··· + a n x n = b variables: x 1 , x 2 , …, x n constants: a 1 , a 2 , …, a n , b real numbers s 1 , s 2 , …, s n If the equation is satisfied, x 1 = s 1 , x 2 = s 2 , …, x n = s n a solution of the linear equation A linear equation has (infinitely) many solutions The set of all solutions: solution set An expression that give us all solutions: general solution What is a general solution of a LE?
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lecture02 - Lecture 2 1.1 Linear Systems and their...

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