3 MatricesDefinitionAn equation such as x+3y=9 is called a linear equation. The graph of this equation is a straight line in the x-y plane.A pair of values of x and y that satisfy the equation is called a solution.
4 DefinitionA linear equation in n variables x1, x2, x3, …, xn has the forma1 x1 + a2 x2 + a3 x3 + … + an xn = bwhere the coefficients a1, a2, a3, …, an and b are real numbers.
5 Solutions for System of Linear Equations Figure 1.2No solution–2x + y = 3–4x + 2y = 2Lines are parallel. No point of intersection. No solutions.Figure 1.3Many solution)4x – 2y = 66x – 3y = 9Both equations have the same graph. Any point on the graph is a solution. Many solutions.Figure 1.1Unique solutionx + 3y = 9–2x + y = –4Lines intersect at (3, 2)Unique solution: x = 3, y = 2.
6 A linear equation in three variables corresponds to a plane in three-dimensional space. ※ Systems of three linear equations in three variables:Unique solution( one solution)
15 Row Echelon Form Definition A matrix is in row echelon form if Any rows consisting entirely of zeros are grouped at the bottom of the matrix.The first nonzero element of each other row is 1. This element is called a leading 1.The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
16 Examples for reduced echelon form ()()()()elementary row operations are used to put a matrix in the row echelon form.
17 Elementary Row Operations of Matrices Elementary TransformationInterchange two equations.2. Multiply both sides of an equation by a nonzero constant.3. Add a multiple of one equation to another equation.Elementary Row OperationInterchange two rows of a matrix.Multiply the elements of a row by a nonzero constant.Add a multiple of the elements of one row to the corresponding elements of another row.
18 Example 2Use the elementary row operations to find the row echelon form of the following matrix.Solutionpivot leading 1)pivotpivotThe matrix is the row echelon form of the given matrix.
19 Solving Linear Systems by Gaussian Elimination Method System of linear equations form augmented matrix put the augmented matrix in row echelon from solve by back substitution
20 Matrix form of a system of linear equations: =Axb
22 Example 3 Solving the following system of linear equation. Solution row equivalentSolutionEquation MethodInitial system:Analogous Matrix MethodAugmented matrix:Eq2+(–2)Eq1Eq3+(–1)Eq1R2+(–2)R1R3+(–1)R1
23 Eq3+(2)Eq2R3+(2)R2(–1/5)Eq3Back substitution(–1/5)R3The solution isThe solution is
24 Example 4Solving the following system of linear equation.Solution
29 Example 9This example illustrates a system that has no solution. Let us try to solve the systemSolution0x1+0x2+0x3=1The system has no solution.
30 Homogeneous System of linear Equations Note. trivial solutionThe system has other nontrivial solutions.Example:TheoremA system of homogeneous linear equations that has more variables than equations has many solutions.
31 Summary If , then the system is independent If , then the system is dependentIf, then the system is inconsistent
32 ExercisesEx 55: Find all values of a for which the following system has a unique solution
33 Ex :Find the interpolating polynomial that passes through the points ( -3,28), (-1,6) and (2,3)