Arithmetic and Algebra of Matrices

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1 Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana

2 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational Numbers 5 The Real Numbers 6 Systems of Linear Equations 7 Representing Linear Systems Using Matrices 8 Polynomial Curve Fitting

3 The Real Numbers The set of irrational numbers together with the set of the rational numbers comprises the set of real numbers. The set of real numbers is usually denoted R

4 The Real Numbers Any number that can be represented in form a b where a and b are integers and b 0 is called rational. The set of all rational numbers is usually denoted Q.

5 The Real Numbers Rational Number Property 1 Different than the integers, the rational numbers are infinitely dense. That is, between any two rational numbers there is an infinite number of rational numbers.

6 The Real Numbers Rational Number Property 2 A number is rational if and only if its decimal representation either terminates or repeats.

7 The Real Numbers Any number that cannot be represented in form a b b are integers and b 0 is called irrational. where a and

8 The Real Numbers How do we know that? 2 is irrational?

9 The Real Numbers TRUE or FALSE: Between any two distinct rational numbers an irrational number can be found?

10 The Real Numbers TRUE or FALSE: Between any two distinct irrational numbers a rational number can be found?

11 Classroom Connection: Systems of Linear Equations

12 Classroom Connection: Systems of Linear Equations

13 Classroom Connection: Systems of Linear Equations

14 Classroom Connection: Systems of Linear Equations

15 Rational Numbers Any number that can be represented in form a b where a and b are integers and b 0 is called rational. The set of all rational numbers is usually denoted Q.

16 Rational Numbers Rational Number Property 1 Different than the integers, the rational numbers are infinitely dense. That is, between any two rational numbers there is an infinite number of rational numbers.

17 Rational Numbers Rational Number Property 2 A number is rational if and only if its decimal representation either terminates or repeats.

18 Irrational Numbers Any number that cannot be represented in form a b where a and b are integers and b 0 is called irrational. The set of all rational numbers is usually denoted Q.

19 Irrational Numbers How do we know that? 2 is irrational?

20 Irrational Numbers TRUE or FALSE: Between any two distinct rational numbers an irrational number can be found?

21 Irrational Numbers TRUE or FALSE: Between any two distinct irrational numbers a rational number can be found?

22 The Real Numbers The set of irrational numbers together with the set of the rational numbers comprises the set of real numbers. The set of real numbers is usually denoted R

23 Systems of Linear Equations What system of linear equations represents the information presented below?

24 Systems of Linear Equations What solution methods can we use to solve the system: 4c ` 8p 8 3c ` 10p 7

25 Systems of Linear Equations Elementary Row Operations - Equations 1 Interchange any two equations 2 Replace any equation by a nonzero constant multiple of itself 3 Replace any equation by the sum of that equation and constant multiple of any other equation

26 Representing Linear Systems Using Matrices When using elementary row operations to solve systems of equations it becomes cumbersome to write variables. Crucial to the solution to the system are the coefficients of the variables and the constant terms. The use of matrices (rectangular arrays of real numbers) helps address this problem.

27 Representing Linear Systems Using Matrices System of Equations 4c ` 8p 8 3c ` 10p 7 Ñ Augmented Matrix j

28 Representing Linear Systems Using Matrices Elementary Row Operations - Matrices 1 Interchange any two rows 2 Replace any row by a nonzero constant multiple of itself 3 Replace any row by the sum of that row and constant multiple of any other row

29 Representing Linear Systems Using Matrices Augmented Gauss Jordan Reduced Row Matrix Elimination Echelon Form j Ñ 1 0 3{ {4 j

30 Representing Linear Systems Using Matrices Reduced Row Echelon Form 1 Each row consisting entirely of zeros lies below all rows having nonzero entries 2 The first nonzero entry in each nonzero row is a 1 (called a leading 1) 3 In any successive nonzero rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row 4 If a column in the coefficient matrix contains a 1, then the other entries in that column are zeros

31 Representing Linear Systems Using Matrices Every system of linear equations has either one, none or infinitely many solutions.

32 Representing Linear Systems Using Matrices Characterize each system below as having one, none or infinitely many solutions. A B C 2x y 1 2x y 1 2x y 1 3x ` 2y 12 6x 3y 3 6x 3y 12 How does each system s solution (or lack thereof) present itself as an augmented matrix?

33 Representing Linear Systems Using Matrices Solve the following system by Gauss-Jordan elimination 3x 2y ` 8z 9 2x ` 2y ` z 3 x ` 2y 3z 8

34 Polynomial Curve Fitting Given n points in the plane, no two of which lie on a vertical line, there exists a unique polynomial function of degree less than or equal to n 1 whose graph passes through the specified points. Such a polynomial is called the interpolating polynomial of minimal degree.

35 Polynomial Curve Fitting Find the interpolating polynomial of minimal degree passing through p3, 6q and p8, 9q

36 Polynomial Curve Fitting Find the interpolating polynomial that models the data below. Year Driving Distance Leader Average Distance 1980 Dan Pohl Andy Bean John Daly Let time start at zero in 1980.

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