Alg2 lesson 4-8
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The document discusses writing and solving systems of equations using matrix form. It explains how to determine the coefficient, variable, and constant matrices and write the corresponding matrix equation. It then provides examples of using the matrix equation to solve systems of equations and discusses what to do when the system has no solution.
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This lecture discusses inverse matrices and matrix rank. It defines an inverse matrix as a matrix A-1 such that A-1A = AA-1 = I, where I is the identity matrix. For a matrix to have an inverse, it must be square and have a non-zero determinant. The lecture provides examples of calculating inverse matrices and shows that the inverse can be found using the adjoint matrix and determinant. Properties of inverse matrices include the inverse of products and transposes. Inverse matrices can also be computed using Gaussian elimination to put the matrix in reduced row echelon form. Matrix rank is also introduced.
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This document provides step-by-step instructions for performing various matrix operations in Excel, including entering data into matrices, adding, subtracting, transposing, multiplying by scalars, multiplying matrices, finding inverses, determinants, and using inverse matrices to solve systems of equations. It explains how to organize data into matrices by typing each element into individual cells. It also describes the necessary steps to perform each operation like highlighting the appropriate cells and entering the correct formulas while using Ctrl+Shift+Enter to handle array calculations. The document serves as a guide for understanding matrices and how Excel can be used to both demonstrate matrix concepts and help learn related operations.
Rank, Nullity, and Fundamental Matrix Spaces.pptx
The document discusses rank and nullity of matrices, and the fundamental matrix spaces. It defines rank as the maximum number of linearly independent rows or columns. The row and column ranks of a matrix are equal, and represent the dimension of the row and column spaces. Nullity is defined as the dimension of the null space. Examples are provided to demonstrate calculating the rank and nullity of matrices by finding their reduced row echelon forms. The rank is the number of pivot columns, and the nullity is the number of free variables.
Matrix equations and inverses honors
This document discusses solving systems of linear equations using matrix equations and inverses. It explains that a system of linear equations can be written as a matrix equation of the form AX=B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. It then demonstrates how to solve these matrix equations by multiplying both sides by the inverse of A, such that X = A^-1B. Several examples are worked through to show how to set up and solve systems of linear equations using this matrix inverse approach.
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1150 day 11
Ratios, proportions, and percents can be used to compare quantities and solve problems. A ratio compares two quantities, like the number of squares to circles. Equal ratios form a proportion, which can be used to solve for unknown values. Percents represent a number out of 100. Common percent calculations include finding a percent of a number and converting between fractions, decimals, and percents. The percent proportion states that the percent equals the part divided by the whole quantity.
1150 day 10
This document covers topics related to repeating decimals, irrational numbers, and square roots. It includes examples of writing repeating decimals as fractions, ordering decimals, finding square roots, and using the Pythagorean theorem. Various math problems are presented along with step-by-step solutions to illustrate these concepts.
1150 day 9
1) Decimals represent numbers using place value with decimal points. To write decimals in expanded form, they are broken into place value terms using positive and negative exponents.
2) Fractions can be converted to decimals using division or by writing the fraction as a ratio of two numbers and setting up a proportion to solve for the decimal.
3) Scientific notation is used to write very large or small numbers in a standard form, such as 3.603 x 107. Operations can be done on decimals by lining up the decimal points or moving them with multiplication/division.
1150 day 8
This document provides an overview of rational numbers including:
- Integers and fractions written in the form a/b where a and b are integers and b ≠ 0
- Equivalent fractions represented by the same number
- Ordering and comparing rational numbers
- Converting between improper fractions and mixed numbers
- Basic operations of addition, subtraction, multiplication, and division of rational numbers
- Word problems involving rational numbers
1150 day 7
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
1150 day 6
Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.
1150 day 5
Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
1150 day 4
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
1150 day 3
The document discusses different number systems used throughout history including Hindu-Arabic, Egyptian, Babylonian, Mayan, and Roman systems. It also covers basic concepts of whole numbers such as addition, subtraction, and their properties. Different models are presented to demonstrate whole number operations including set, number line, take-away, comparison, and missing addend models.
1150 day 2
The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.
1150 day 1
The number of terms in each sequence can be determined by finding a one-to-one correspondence with the counting numbers.
Alg2 lesson 13-5
This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Alg2 lesson 13-4
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-2
The document describes how to draw angles in standard position on a unit circle and convert between degrees and radians. It provides examples of drawing angles such as 210°, -45°, and 540° in standard position and rewriting angles such as 30°, 45°, and an unspecified angle in radians and degrees. It also gives examples of finding coterminal angles for 210° with one positive and one negative measure.
Alg2 lesson 13-1
This document discusses trigonometric functions and their ratios, including sine, cosine, tangent, cosecant, secant, and cotangent. It provides examples of using trigonometric functions to solve values in right triangles, including finding missing side lengths and angle measures. Special right triangles with ratios of 1/2, 1/√3, and 1/√2 are also covered.
Alg2 lesson 11-2 and 11-4
The document provides information about arithmetic and geometric series. It defines arithmetic and geometric series, provides examples of finding sums of arithmetic series using formulas, and defines the key terms (first term, common ratio, number of terms, last term) used in the formula to calculate the sum of a geometric series.
Alg2 lesson 11-3
The document discusses geometric sequences, which are patterns of numbers where each term is found by multiplying the previous term by a constant called the common ratio. It provides examples of geometric sequences with different common ratios and shows how to write equations to find the nth term in a sequence. It also explains how to find specific terms in a geometric sequence and calculates the geometric means between two nonconsecutive terms.
Alg2 lesson 11-1
The document discusses arithmetic sequences, which are patterns of numbers where each term is found by adding a constant value to the previous term. It provides examples of arithmetic sequences with different common differences and shows how to write equations to determine the nth term. It also explains how to find terms within an arithmetic sequence and determine arithmetic means between two non-consecutive terms.
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Alg2 lesson 4-8
- 1. A system of equations can be written in matrix form. Determine the coefficient, variable, and constant matrices. Write the matrix equation. A • X B = Answer: Example 8-1a
- 2. Why does it work??? Example 8-1a
- 3. Write a matrix equation for the system of equations. Answer: Example 8-1b
- 4. Use the matrix equation to solve the system of equations. B A X B A-1 A-1 A = X A-1 I = X B A-1 = X B Example 8-1b
- 5. A-1 = X B
- 6. Use a matrix equation to solve the system of equations. A X = B The matrix equation is Example 8-3a
- 7. A X = B = A-1 = X B Answer: The solution is (5, –4). Example 8-3b
- 8. Use a matrix equation to solve the system of equations. The matrix equation is when Example 8-4a
- 9. The determinant of the coefficient matrix is 0,so A–1 does not exist. A-1 = X B Example 8-4b
- 10. Graph the system of equations. Since the lines are parallel, thissystem has no solution. Example 8-4c