Chapter1 - Chapter 1 Linear Systems and Gaussian...

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Chapter 1 Linear Systems and Gaussian Elimination Section 1.1 Linear Systems and Their Solutions Discussion 1.1.1 A line in the xy -plane can be represented algebraically by an equation of the form ax + by = c where a and b are not both zero. An equation of this kind is known as a linear equation in the variables of x and y . In general, we have the following definition. Definition 1.1.2 A linear equation in n variables x 1 , x 2 , . . . , x n has the form a 1 x 1 + a 2 x 2 + ··· + a n x n = b where a 1 , a 2 , . . . , a n and b are real constants. The variables in a linear equation are also called the unknowns . Example 1.1.3 1. The equations x + 3 y = 7, x 1 + 2 x 2 + 2 x 3 + x 4 = x 5 , y = x 1 2 z + 4 . 5 and x 1 + x 2 + ··· + x n = 1 are linear. 2. The equations xy = 2, sin( θ ) + cos( φ ) = 0 . 2, x 2 1 + x 2 2 + ··· + x 2 n = 1 and x = e y are not linear. 3. The linear equation ax + by + cz = d , where a, b, c, d are constants and not all a, b, c are zero, represents a plane in the three dimensional space. For example, z = 0 (i.e. 0 x + 0 y + z = 0) is the xy -plane contained inside the three dimensional space.
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2 Chapter 1. Linear Systems and Gaussian Elimination a45 y a54 z a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a9 x a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 a0 z = 0 Definition 1.1.4 Given n real numbers s 1 , s 2 , . . . , s n , we say that x 1 = s 1 , x 2 = s 2 , . . . , x n = s n is a solution of a linear equation a 1 x 1 + a 2 x 2 + ··· + a n x n = b if the equation is satisfied when we substitute the values into the equation accordingly. The set of all solutions of the equation is called the solution set of the equation and an expression that gives us all these solutions is called the general solution of the equation. Example 1.1.5 1. Consider the linear equation 4 x 2 y = 1. The general solution is braceleftbigg x = t y = 2 t 1 2 where t is an arbitrary parameter. We can also write the general solution as braceleftbigg x = 1 2 s + 1 4 y = s where s is an arbitrary parameter. These include all solutions such as braceleftbigg x = 1 y = 1 . 5 , braceleftbigg x = 1 . 5 y = 2 . 5 , braceleftbigg x = 1 y = 2 . 5 , and infinitely many other solutions. 2. Consider the equation x 1 4 x 2 + 7 x 3 = 5. The general solution is x 1 = 5 + 4 s 7 t x 2 = s x 3 = t where s, t are arbitrary parameters. 3. (Geometrical Interpretation) (a) In the xy -plane, the solutions ( x, y ) = (1 s, s ), for s R , to the equation x + y = 1 are points on the line as shown below.
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Section 1.1. Linear Systems and Their Solutions 3 a45 x a54 y a64 a64 a64 a64 a64 a64 a64 a64 a64 x + y = 1 a115 (1 s,s ) (b) In the three dimensional space, the solutions to the equation x + y = 1 (i.e. x + y + 0 z = 1) are ( x, y, z ) = (1 s, s, t ) for s, t R . These points form a plane as shown below. a45 y a54 z a0 a0 a0 a0 a0 a0 a0 a9 x a33a33 a33a33 a33a33 a33a33 a33 a33 a33a33 a33a33 a33a33 a33a33 a33 a33 a33a33 a33a33 a33a33 a33a33 a33 a33 x + y = 1 a114 (1 s,s, 0) a114 (1 s,s,t ) Definition 1.1.6 A finite set of linear equations in the variables x 1 , x 2 , . . . , x n is called a system of linear equations (or a linear system ): a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 .
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