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**Unformatted text preview: **___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 2-1 Chapter 2. Systems of linear equations and matrices Linear equation c by ax = + , d cz by ax = + + Definition A linear equation in the n-variables { x 1 , x 2 , …, x n } is an equation that can be written in the form, b x a x a x a n n = + + + L 2 2 1 1 , where the coefficients { a 1 , a 2 , …, a n } and the constant term b are constant. Non-linear equations contain products, reciprocals, or the other functions of the variables, whereas, in linear equations, the variables occur only to the first power and are multiplied only by constants. Examples of non-linear equations are; 1 2 = + z xy , 3 2 2 = − y x , 1 ) 2 sin( 4 2 = − + z y x π , … A system of linear equations is a finite set of linear equations, each with the same variables. A solution of a system of linear equations is a vector that is simultaneously a solution of each equation in the system. Solution to a linear equation Given a system of linear equations, we are concerned with its solutions in terms of (1) Does solution exist? (consistent vs. inconsistent) (2) Is solution unique? (3) What is the geometrical meaning of the solution? (4) What is its meaning in the vector space? A system of linear equations is called consistent if it has at least one solution (otherwise, inconsistent ). If we have two lines in the xy-plane, there relationship can belong to one of the following three cases; two lines intersecting at one point (a unique solution), two parallel lines (no solution), and two identical lines (infinitely many solutions). Similarly, there are three possibilities for any system of linear equations, (1) a unique solution (a consistent system), (2) infinitely many solutions (a consistent system), (3) no solution (an inconsistent system). Consider the following examples; (a) 1 2 4 = − y x (one equation with two unknowns: a line in the xy-plane) Choose a parameter t , set t x = and solve for y to get 2 / 1 2 − = t y . If we choose t y = , then 4 / 1 2 / + = t x : This implies that there exist a infinitely many solutions. ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 2-2 (b) 1 2 = − y x ; 5 = + y x (two equations with two unknowns: two lines in the xy-plane: solution corresponds to the intersection of two lines) Using a matrix notation, ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 5 1 1 1 1 2 y x or ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 5 1 1 1 1 2 y x Note that this matrix-vector equation is a linear combination of two vectors (2, 1) and (-1, 1) with coefficients x and y to construct a vector (1, 5). Consider the set of all possible to construct a vector (1, 5)....

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