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Chapter 1

Chapter 1. Section 1.1 Introduction to Matrices and Systems of Linear Equations. This is called a linear combination of the variables . Linear Combinations A linear combination of the n variables is an expression of the form given to the right where are known constants (numbers).

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Chapter 1

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  1. Chapter 1 Section 1.1 Introduction to Matrices and Systems of Linear Equations

  2. This is called a linear combination of the variables Linear Combinations A linear combination of the n variables is an expression of the form given to the right where are known constants (numbers) Is a linear combination of x and y Is a linear combination of Linear Equations An equation with n different variables is called a linear equation if it is possible it write the equation (maybe using some equivalent algebraic rearrangement) as a linear combination where are called the coefficients of the equation being set equal to a constant b. Nonlinear Equations Linear Equation Or Linear Equation 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.

  3. Systems of linear equations and their solution A system of linear equations is a set of mlinear equations with n variables. The numbers represent the coefficient of the jth variable in the ith equation. Such a system can be expressed in a form given to the below. A system of linear equations. A solution to a system of equations (sometimes called a simultaneous solution) with n variables is a set of numbers such that all the numbers satisfy all of the m equations in the system. A solution to a system of linear equations.

  4. Linear Combinations of functions Let be a set of n functions (not necessarily linear) and a set of constants we can form a linear combination of functions as shown to the right. A linear combination of functions A linear combination of the functions 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: Nonlinear System New System: Linear System Substitute:

  5. 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. Look at the previous example: The coordinates of the point the lines cross are (2,5) slope is 3/2 y-intercept is 2 slope is -4 y-intercept is 13 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.

  6. More graphing examples: Solution (-3,2) x = -3 Vertical line at -3 y = 2 Horizontal line at 2 More graphing examples: slope is 3 y-intercept is -4 slope is 3 y-intercept is -1 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.

  7. 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. system of equations system of equations Augmented Matrix Augmented Matrix Sometimes algebra might be needed to change the equations to a matrix.

  8. 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. A system of linear equations. The augmented matrix for the system 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: Where b is the matrix to the right.

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