system of linear equations nattee niparnan. linear equations
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Linear Equation
• An Equation– Represent a straight line– Is a “linear equation” in the variable x and y.
• General form– ai a real number that is a coefficient of xi
– b another number called a constant term
System of a Linear Equation
• A collection of several linear equations– In the same variables
• What about– A linear equation• in the variables x1, x2 and x3
– Another equation• in the variables x1, x2,x3 and x4
– Do they form a system of linear equation?
Solution
• A linear equation
• Has a solution
• When
• It is called a solution to the system if it is a solution to all equations in the system
Parametric Form
• Solution of the system in Equation 1 is described in a parametric form– It is given as a function in parameters s and t– It is called a general solution of the system
• Every linear equation system having solutions– Can be written in parametric form
Try another one
• Solve it using parametric form
• In term of x and z
• In term of y and z
There are several general
solutions
Geometrical Point of View
• In the case of 2 variables– Each equation is represent a line in 2D– Every point in the line satisfies the equation
• If we have 2 equations– 3 possibilities• Intersect in a point• Intersect as a line• Parallel but not intersect
Higher Space?
• Somewhat difficult to imagine– But Linear Algebra will, at least, provides some
characteristic for us
Cogito, ergo sum
I also speak Calculus
Equivalent System
• System a set of linear equations– Two systems having the same
solution is said to be “equivalent”
• Some system is easier to identify the solution
• To solve a system, we manipulate it into an “easy” system that is still equivalent to the original system
System 1
System 2
System 3
Solution preserve operation
Solution preserve operation
Elementary Operation
• Interchange two equations• Multiply one equation with a nonzero number• Add a multiple of one equation to a different
equation
Theorem 1
• Suppose that an elementary operation is performed on a linear equation system– Then, there solution are still the same
Elementary Row Operation
• We don’t really do the elementary operation• We write the system as an augmented matrix
and then perform “elementary row operation” on that matrix
Gaussian Elimination
• An algorithm that manipulate an augmented matrix into a “nice” augmented matrix
Row Echelon Form
• A matrix is in “Row Echelon Form” (called row echelon matrix) if– All zero rows are at the bottom– The first nonzero entry from the left in each
nonzero row is 1 • (that 1 is called a leading 1 of that row)
– Each leading 1 is to the right of all leading 1’s in the row above it
Reduced Row Echelon
• The leading 1 is the only nonzero element in that column
row echelon
Reduced row echelon
Theorem 2
• Every matrix can be manipulated into a (reduced) row echelon form by a series of elementary row operations
Solution to (c)
Variable corresponding to the leading 1’s is called “leading variable”
The non-leading variables end up as a parameter in the solution
Gaussian Elimination
• If the matrix is all zeroes stop• Find the first column from the left containing a
non zero entry (called it A) and move the row having that entry to the top row
• Multiply that row by 1/A to create a leading 1• Subtract multiples of that row from rows below
it, making entry in that column to become zero• Repeat the same step from the matrix consists of
remaining row
Redundancy
Subtract 2 time row 1 from row 2AndSubtract 7 time row 1 from row 3
Subtract 2 time row 2 from row 1AndSubtract 3 time row 2 from row 3
Back Substitution
• Gaussian Elimination brings the matrix into a row echelon form– To create a reduced row echelon form• We need to change step 4 such that it also create zero
on the “above” row as well• Usually, that is less efficient
• It is better to start from the row echelon form and then use the leading 1 of the bottom-most row to create zero
Rank
• It is (later) shown that, for any matrix A, it has the same “Reduced row echelon form”– Regardless of the elementary row operation performed
• But it s not true for “row echelon form”– Different sequence of operations leads to different row
echelon matrix• However, the number of leading 1’s is always the same
– Will be proved later• Hence, the number of leading 1’s depends on A• The number of leading 1’s is called rank of A
Theorem 3
• Suppose a system of m equation on n variables has a solution, if the rank of the augmented matrix is r – the set of the solution involve exactly n-r
parameters
Homogeneous Linear System
• Xi = 0 is always a solution to the homogeneous system– It is called “trivial” solution
• Any solution having nonzero term is called “nontrivial” solution
Existence of Nontrivial Solution to the homogeneous system
• If it has non-leading entry in the row echelon form– The solution can be described as a parameter
• Then it has nonzero solution!!!– Nontrivial
• When will we have non-leading entry?– When we have more variable than equation
Network Flow Problem
• A graph of traffic– Node = intersection– Edge = road– Do we know the flow at each road?