Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of mathematical notation wikipedia , lookup
Braโket notation wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Line (geometry) wikipedia , lookup
Elementary algebra wikipedia , lookup
Recurrence relation wikipedia , lookup
System of polynomial equations wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Linear algebra wikipedia , lookup
History of algebra wikipedia , lookup
Chapter 1 Section 1.1 Introduction to Matrices and Systems of Linear Equations Linear Combinations A linear combination of the n variables ๐ฅ1 , ๐ฅ2 , ๐ฅ3 , โฏ , ๐ฅ๐ is an expression of the form given to the right where ๐1 , ๐2 , ๐3 , โฏ , ๐๐ are known constants (numbers) ๐1 ๐ฅ1 + ๐2 ๐ฅ2 + ๐3 ๐ฅ3 + โฏ + ๐๐ ๐ฅ๐ This is called a linear combination of the variables ๐ฅ1 , ๐ฅ2 , ๐ฅ3 , โฏ , ๐ฅ๐ . 2๐ฅ + 3๐ฆ Is a linear combination of x and y 2๐ฅ1 + 4๐ฅ4 + 5๐ฅ6 Is a linear combination of ๐ฅ1 , ๐ฅ4 , ๐ฅ6 Linear Equations An equation with n different variables ๐ฅ1 , ๐ฅ2 , ๐ฅ3 , โฏ , ๐ฅ๐ is called a linear equation if it is possible it write the equation (maybe using some equivalent algebraic rearrangement) as a linear combination where ๐1 , ๐2 , ๐3 , โฏ , ๐๐ are called the coefficients of the equation being set equal to a constant b. ๐1 ๐ฅ1 + ๐2 ๐ฅ2 + ๐3 ๐ฅ3 + โฏ + ๐๐ ๐ฅ๐ = ๐ 7๐ฅ + 2๐ฆ โ 3๐ง = 9 Linear Equation 4๐ฅ1 โ 2๐ฅ4 = 3 2 โ ๐ฅ2 + 5๐ฅ4 Or 4๐ฅ1 + 3๐ฅ2 โ 17๐ฅ4 = 6 Linear Equation ๐ฅ2 ๐ฅ5 + ๐ฅ1 = 6 3 cos ๐ผ โ 2 sin ๐ฝ = 8 Nonlinear Equations If the equation (maybe after some algebra) is anything other than a number times a variable being added or subtracted it is nonlinear. No powers, roots, variables in the denominator, products of variables, trig functions etc. Systems of linear equations and their solution A ๐ × ๐ system of linear equations is a set of m linear equations with n variables. The numbers ๐๐๐ represent the coefficient of the jth variable in the ith equation. Such a system can be expressed in a form given to the below. ๐11 ๐ฅ1 ๐21 ๐ฅ1 โฎ ๐๐1 ๐ฅ1 + + + ๐12 ๐ฅ2 ๐22 ๐ฅ2 โฎ ๐๐2 ๐ฅ2 + + โฏ โฏ + + + โฏ + ๐1๐ ๐ฅ๐ ๐2๐ ๐ฅ๐ โฎ ๐๐๐ ๐ฅ๐ = = = ๐1 ๐2 โฎ ๐๐ A ๐ × ๐ system of linear equations. A solution to a system of equations (sometimes called a simultaneous solution) with n variables is a set of numbers ๐ 1 , ๐ 2 , ๐ 3 , โฏ , ๐ ๐ such that all the numbers satisfy all of the m equations in the system. ๐11 ๐ 1 ๐21 ๐ 1 โฎ ๐๐1 ๐ 1 + + + ๐12 ๐ 2 ๐22 ๐ 2 โฎ ๐๐2 ๐ 2 + + โฏ โฏ + + + โฏ + ๐1๐ ๐ ๐ ๐2๐ ๐ ๐ โฎ ๐๐๐ ๐ ๐ = = = ๐1 ๐2 โฎ ๐๐ A solution to a ๐ × ๐ system of linear equations. Linear Combinations of functions Let ๐1 ๐ฅ , ๐2 ๐ฅ , โฏ , ๐๐ ๐ฅ be a set of n functions (not necessarily linear) and ๐1 , ๐2 , โฏ , ๐๐ a set of constants we can form a linear combination of functions as shown to the right. ๐1 ๐1 ๐ฅ + ๐2 ๐2 ๐ฅ + โฏ + ๐๐ ๐๐ ๐ฅ A linear combination of functions ๐1 ๐ฅ , ๐2 ๐ฅ , โฏ , ๐๐ ๐ฅ 2๐ฅ 2 + 3 cos ๐ฅ + 5 ๐ฅ A linear combination of the functions ๐ฅ 2 , cos ๐ฅ , ๐ฅ Substitution to a Linear System If all of the equations in a nonlinear system are linear combinations of the same functions a substitution can be done to transform the nonlinear system into a linear system. System: ๐ฅ + 4 sin ๐ฆ = 5 3 ๐ฅ โ 2 sin ๐ฆ = 8 Nonlinear System Substitute: ๐ฅ1 = ๐ฅ ๐ฅ2 = sin ๐ฆ New System: ๐ฅ1 + 4๐ฅ2 = 5 3๐ฅ1 โ 2๐ฅ2 = 8 Linear System Solving by Graphing As the name would imply the graphs of linear equations are lines. The idea is to graph both lines on the same graph carefully. Look at the point where the two lines cross (or try to estimate it as best as you can) the x and y coordinates are the simultaneous solutions to the system of equations. ๏ฌ๏ญ 3 x ๏ซ 2 y ๏ฝ 4 Look at the previous example: ๏ญ ๏ฎ 4 x ๏ซ y ๏ฝ 13 15 14 13 12 ๏ญ 3x ๏ซ 2 y ๏ฝ 4 2 y ๏ฝ 3x ๏ซ 4 11 4 x ๏ซ y ๏ฝ 13 y ๏ฝ ๏ญ4 x ๏ซ 13 10 9 8 7 y ๏ฝ x๏ซ2 3 2 6 The coordinates of the point the lines cross are (2,5) 5 4 slope is 3/2 slope is -4 3 2 y-intercept is 2 y-intercept is 13 1 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -1 The problem that you run into with graphing to find the solutions is that it can be very imprecise. When the solutions involve fractions or more than 2 or 3 variables this is very imprecise and not practical. This is why we will look at other algebraic methods that tell you the simultaneous solutions. 15 ๏ฌ x ๏ฝ ๏ญ3 ๏ญ ๏ฎy ๏ฝ 2 More graphing examples: 5 4 3 2 Solution (-3,2) x = -3 Vertical line at -3 y=2 -5 -4 1 -3 -2 -1 1 2 3 4 5 1 2 3 4 5 -1 -2 Horizontal line at 2 -3 -4 -5 5 ๏ฌ๏ญ 6 x ๏ซ 2 y ๏ฝ ๏ญ8 ๏ฎ 3x ๏ญ y ๏ฝ 1 4 More graphing examples: ๏ญ ๏ญ 6 x ๏ซ 2 y ๏ฝ ๏ญ8 2 y ๏ฝ 6x ๏ญ 8 3x ๏ญ y ๏ฝ 1 ๏ญ y ๏ฝ ๏ญ3 x ๏ซ 1 y ๏ฝ 3x ๏ญ 4 y ๏ฝ 3x ๏ญ 1 3 2 1 -5 -4 -3 -2 -1 -1 -2 slope is 3 slope is 3 y-intercept is -4 y-intercept is -1 -3 -4 -5 These lines are parallel which means they do not intersect. This means there is no simultaneous solution to the system of equations. A system of equations that has no simultaneous solution we call inconsistent. Systems of Equations and Augmented Matrices Systems of equations can be represented with matrices in a certain way. 1. Each row corresponds to an equation. 2. Each column to a variable and the last column to the constants. We write the variables on one side of the equation and the constants on the other. In the matrix separate the variables from the constants with a line (sometimes dashed). The entries of the matrix are the coefficients of the variables. It is important that if a variable does not show up in an equation that means the coefficient is 0 and that entry in the matrix is 0. The entries on the other side of the line are the constants. ๏ฌ3x ๏ญ 4 y ๏ฝ 7 ๏ญ ๏ฎ9 x ๏ซ y ๏ฝ 8 system of equations Sometimes algebra might be needed to change the equations to a matrix. ๏ฉ3 ๏ญ 4 7๏น ๏ช ๏บ 9 1 8 ๏ซ ๏ป ๏ฌx ๏ซ 3 y ๏ญ 5z ๏ฝ 8 ๏ฏ ๏ญ x๏ญz ๏ฝ9 ๏ฏ 4 y ๏ซ 3z ๏ฝ 1 ๏ฎ Augmented Matrix ๏ฌ y ๏ฝ 3x ๏ญ 2 ๏ญ ๏ฎ2( x ๏ซ 3) ๏ฝ 5 y ๏ญ 10 ๏ฌ๏ญ 3x ๏ซ y ๏ฝ ๏ญ2 ๏ญ ๏ฎ2 x ๏ซ 6 ๏ฝ 5 y ๏ญ 10 system of equations ๏ฌ๏ญ 3x ๏ซ y ๏ฝ ๏ญ2 ๏ญ ๏ฎ2 x ๏ญ 5 y ๏ฝ ๏ญ16 ๏ฉ1 3 ๏ญ 5 8 ๏น ๏ช ๏บ 1 0 ๏ญ 1 9 ๏ช ๏บ ๏ช๏ซ0 4 3 1๏บ๏ป Augmented Matrix ๏ฉ๏ญ 3 1 ๏ญ 2 ๏น ๏ช ๏บ 2 ๏ญ 5 ๏ญ 16 ๏ซ ๏ป Matrix Representation and Notation The augmented matrix is one matrix associated with the system of equations. There is another matrix which we refer to as the coefficient matrix. ๐ด = ๐๐๐ ๐11 ๐12 = โฎ ๐๐1 ๐12 ๐22 โฎ ๐๐2 โฏ โฏ โฏ ๐11 ๐ฅ1 ๐21 ๐ฅ1 โฎ ๐๐1 ๐ฅ1 ๐1๐ ๐2๐ โฎ ๐๐๐ The coefficient matrix for the system The matrix B is the coefficient matrix A "augmented" with the column of constants. This is sometimes written as: ๐ด๐ Where b is the matrix to the right. + + + ๐12 ๐ฅ2 ๐22 ๐ฅ2 โฎ ๐๐2 ๐ฅ2 + + โฏ โฏ + + + โฏ + ๐1๐ ๐ฅ๐ ๐2๐ ๐ฅ๐ โฎ ๐๐๐ ๐ฅ๐ = = = A ๐ × ๐ system of linear equations. ๐11 ๐12 ๐ต= โฎ ๐๐1 ๐12 ๐22 โฎ ๐๐2 โฏ โฏ โฏ ๐1๐ ๐1 ๐2๐ ๐2 โฎ โฎ ๐๐๐ ๐๐ The augmented matrix for the system ๐1 ๐ ๐= 2 โฎ ๐๐ ๐1 ๐2 โฎ ๐๐