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Unformatted text preview: rs. Outline Linear Systems Matrices Examples Example
Solve the linear equations.
1 3x + 6y = 20 2 x − 4y + 9z = 5 Gaussian Elimination Homogeneous Systems Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Linear Systems of Equations
A collection of m linear equations in n unknowns is called a
system of linear equations.
We can write the system as:
a11 x1 +
a21 x1 +
. a12 x2 + · · ·
a22 x2 + · · ·
. am1 x1 + am2 x2 + · · · +
+ a1n xn =
a2n xn =
. (1) + amn xn = bm The real number aij is the coeﬃcient of xj in the i th equation.
The real number bi is the constant term in the i th equation.
A solution of the system is an ordered list of numbers
s1 , s2 , . . . , sn that is a solution of each equation in the system.
To solve the system means to ﬁnd all solutions of the system. Outline Linear Systems Matrices Gaussian Elimination Examples Example
Consider the system of two equations in two unknowns:
1 Is (2, 3) a solution? 2 Is (5, 5) a solution? − 3y
+ 4y = −5
= 10 Homogeneous Systems Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Solving Linear Systems of Equations Two systems of equations which have the same solution set
are said to be equivalent systems.
One approach to solving a system of linear equations is to
transform it into an equivalent system whose solutions are
obvious. For example:
.. . = s1
x n = sn (2) Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Operations That Respect Solutions
Suppose (s1 , s2 , . . . , sn ) is a solution of both linear equations:
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b1
(s1 , s2 , . . . , sn ) is also a solution of the sum of the two
(a11 + a21 )x1 + (a12 + a22 )x2 + · · · + (a1n + a2n )xn = b1 + b2 .
(s1 , s2 , . . . , sn ) is solution of a constant multiple λ of either
λa11 x1 + λa12 x2 + · ·...
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