Chapter 1- Linear Equations in Linear Algebra
Topics of Study
Chapter 1 Linear Equations in Linear Algebra / Chapter 4 Vector Spaces
1.1 Systems of Linear Equations / 4.1 Vector Spaces and Subspaces
1.2 Row Reduction and Echelon Forms / 4.2 Null Spaces, Column Spaces, and Linear Transformations
1.3 Vector Equations / 4.3 Linearly Independent Sets; Bases
1.4 The Matrix Equation Ax = b / 4.4 Coordinate Systems
1.5 Solution Sets of Linear Systems / 4.5 The Dimension of a Vector Space
1.6 Applications of Linear Systems* / 4.6 Rank
1.7 Linear Independence / 4.7 Change of Basis
1.8 Introduction to Linear Transformations / 4.8 Applications to Difference Equations*
1.9 The Matrix of a Linear Transformation / 4.9 Applications to Markov Chains*
1.10 Linear Models in Business, Science, and Engineering
Chapter 5 Eigenvalues and Eigenvectors
Chapter 2 Matrix Algebra / 5.1 Eigenvectors and Eigenvalues
2.1 Matrix Operations / 5.2 The Characteristic Equation
2.2 The Inverse of a Matrix / 5.3 Diagonalization
2.3 Characterizations of Invertible Matrices / 5.4 Eigenvectors and Linear Transformations
2.5 Matrix Factorizations / 5.5 Complex Eigenvalues
2.6 The Leontief Input-Output Model* / 5.6 Discrete Dynamical Systems*
5.7 Applications to Differential Equations*
Chapter 3 Determinants
3.1 Introduction of Determinants / Chapter 6 Orthogonality and Least Squares
3.2 Properties of Determinants
3.3 Cramer’s Rule, Volume, and Linear Transformations
Linear Algebra Name:______
Lesson- Systems Graphically
Date:______
Objectives: · solve systems of equations graphically
· determine whether a system of linear equations is consistent and independent, consistent and dependent, or inconsistent
Classifying Systems of Equations:
Consistent System: / Inconsistent System:Independent: / Dependent:
Without graphing, state whether each of the following systems are consistent and independent, consistent and dependent, or inconsistent. Explain your reasoning.
2y + 3x = 64y = 16 – 6x / (1) 40y – 35x = 55
7x = 8y – 11
Solve the following systems of equations graphically:
(2) y = 3x – 2
x + y = 6
Solve the following systems of equations algebraically:
(3) x + 4y = 26
x – 5y = -10
(4) 2x + 3y = 12
5x – 2y = 11
(5) -3x + 5y = 12
6x – 10y = -21
Linear Algebra Name:______
Lesson- Linear Systems Applications- Graphical
Date:______
Objective: · solve real-world applications graphically with systems of equations
(1) Wayne started with 50 gallons of water in his pool, and he is filling it at a rate of 10 gallons per minute. His next-door neighbor Ted started with 20 gallons of water in his pool, and he is filling it at a rate of 15 gallons per minute. Which system of equations could you use to find when the pools will contain the same amount of water?
(A) y = 50 + 15xy = 20 + 10x / (B) y = 50 + 10x
y = 20 + 15x / (C) y = 50 – 15x
y = 20 – 10x / (D) y = 50 – 10x
y = 20 – 15x
(2) Cramer’s Café is a local eatery that plans on selling copies of their famous recipes in a cookbook to raise funds for renovations. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will be sold for $6 each.
(a) Write equations that represent the cost and income from these cookbooks.
(b) Graph the equations.
(c) Use your graph from part (b) to determine how many cookbooks must be sold before Cramer’s Café makes a profit.
(3) Teknotronics, Inc. needs to buy new software for their office computers. The Premium package costs $13,000 plus $500 for each additional site license. The Deluxe package costs $2,500 plus $1,200 for each additional site license.
(a) Write equations that represent the cost of each software package.
(b) Graph the equations.
(c) Use your graph from part (b) to estimate the break-even point of the software costs.
(d) If Teknotronics, Inc. plans to buy 10 additional site licenses, which software will cost less?
(4) Megan is thinking about leasing a car for two years. The dealership says that they will lease her the car she has chosen for $326 per month with only $200 down. However, if she pays $1600 down, the lease payment drops to $226 per month.
(a) Write equations that represent the amount Megan will pay with each plan.
(b) Graph the equations.
(c) Use your graph from part (b) to determine the break-even point when comparing these lease options.
(d) Which 2-year lease should she choose if the down payment is not a problem? Explain your answer.
(5) A tub containing 16 gallons of water is draining at a rate of 1 gallon per hour. A basin of 3.5 gallons of water is draining at a rate of 1 gallon every 6 hours.
(a) Write a system of equations that represent y, the number of gallons left in the container after x hours.
(b) Graph the equations.
(c) If both containers began draining at the same time, how soon will the tub and basin hold the same amount of water?
(d) When the amounts are equal, how much water will be in each container?
(6) Paloma has $1500 in a savings account. She adds $30 to her account each month. Jose has $2400 in his savings account. He withdraws $30 from his account each month.
(a) Write a system of equations that represents this situation.
(b) Graph the equations.
(c) In how many months will they have the same balance in their savings accounts?
(d) What will be the balance in each account?
Linear Algebra Name:______
Lesson- Linear Systems Applications- Algebraic
Date:______
Objectives: · solve real-world applications algebraically with systems of equations
(1) In January, Paloma’s long-distance bill was $5.50 for 25 minutes of calls. In February, her bill was $6.54 for 38 minutes of calls. Find the flat rate and charge per minute for long distance that the phone company is charging Paloma.
(2) Last year, the high school rugby team bought team shirts for $17 each and socks for $5. The total purchase was $315. This year, they bought the same number of items, but the total purchase was $342. This was because the shirts were now being sold for $18 and the socks for $6.
(a) Write a system of equations that represents the number of shirts and socks bought each year.
(b) Using your answer from part (a), algebraically find the number of shirts and socks the team bought each year.
(3) A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered.
(a) Write a system of equations that describes the above situation.
(b) Using your answer from part (a), find the total number of adults and the total number of children in the group.
(4) A mail order company charges for postage and handling according to the weight of the package. A light package is charged $2 and a heavy package is charged $3. An order of 12 packages had a total postage and handling charge of $29. Write and solve a system of equations to determine how many light packages and how many heavy packages were in the order.
(5) The Polynomial Park’s Recreation Department ordered a total of 100 baseballs and bats for the summer baseball camp. Each baseball costs $4.50, and the bats cost $20.00 each. The total purchase price cost $822. Write and solve a system of equations to find how many of each item was ordered.
(6) Mr. McAlley is writing a test for his history class. The test will have true and false questions worth two points each and multiple choice questions worth four points each, for a total of 100 points. He wants to have twice as many multiple choice questions as true and false questions.
(a) Write a system of equations that represents the number of each type of question.
(b) Use your answer from part (a) to find how many of each type of question will be on the test.
(c) If most of his students can answer true and false questions within 1 minute and multiple choice questions within 1½ minutes, will they have enough time to finish the test in 45 minutes?
(7) Tyrone is responsible for checking a shipment of technology equipment that contains laser printers that cost $700 each and color monitors that cost $200 each. He counts 30 boxes on the loading dock. The invoice states that the order totals $15,000.
(a) Write a system of two equations that represents the number of each item.
(b) Use your answer from part (a) to find how many laser printers and how many color monitors were delivered.
(8) Megan exercises every morning for 40 minutes. She does a combination of step aerobics, which burns about 11 Calories per minute, and stretching, which burns about 4 Calories per minute. Her goal is to burn 335 Calories during her routine.
(a) Write a system of equations that represents Megan’s morning workout.
(b) Use your answer from part (a) to find how long she should participate in each activity in order to burn 335 Calories.
Linear Algebra Name:______
Lesson- Systems of Linear Equations- Matrices (1.1)
Date:______
Objectives: · solve a system of linear equations using elementary row operations.
Definition of a Linear Equation in n Variables:
A linear equation in n variable has the form
,
where the coefficients are real numbers (usually known). The number of is the leading coefficient and is the leading variable.
In this section, we are interested in the collection of several linear equations. The collection of these linear equations are referred to as a system of linear equations.
Definition of System of m Linear Equation in n Variables:
A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables:
where are constants.
Example:
Consider the following system of linear equations:
In matrix form it would look like this:
Gaussian Elimination
Matrices can be used to represent systems of equations, which we then try to solve. We would do this by manipulating the matrix into a simpler form. We can do this using some Elementary Matrix Operations.
There are three kinds of elementary matrix operations.
1. Interchange two rows.
2. Multiply each element in a row by a non-zero number.
3. Multiply a row by a non-zero number and add the result to another row
Example 1:
Solve the following system using Gaussian elimination.
1x + 2y = 1
3x + 7y = 2
Solution:
The goal is to reduce the matrix to something easy to work with, namely something with all zeros below the diagonal.
What did I do?
à
à
Using the reduced form we can now determine that the solution is as follows:
Example 2:
In matrix form:
Solution:
à
à
à
à
à
à
Exercises:
1. 3x – 2y = 1
2x – 3y = 9
2. 3x + 2y = 5
4x + 5y = 7
3.HW- read 1.1 do p10 #1-21 every other odd
Linear Algebra Name:______
Lesson- Row Reduction and Echelon Forms (1.2)
Date:______
Objectives: · solve a system of linear equations using elementary row operations.
Special Matrices:
1. A square matrix is a matrix with the same number of rows as columns.
2. A diagonal matrix is a square matrix whose entries off the main diagonal are zero.
3. An upper triangular matrix is a matrix having all the entries below the main diagonal equal to zero.
4. A lower triangular matrix is a matrix having the entries above the main diagonal equal to zero.
Row Reduction and Echelon Forms (rref)
DEFINITION 1: A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry.
DEFINITION 2: A leading entry of a row is the leftmost nonzero entry in a nonzero row.
DEFINITION 3: A rectangular matrix is in echelon form if it has the following properties:
(1) All nonzero rows are above any rows of all zeros.
(2) Each leading entry is in a column to the right of the leading entry of the row above it.
(3) All entries in a column below the leading entry are all zeros.
DEFINITION 4: If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form:
(4) The leading entry in each nonzero row is 1
(5) Each leading 1 is the only nonzero entry in its column.
Note: The instance of using the rref to solve a system is known as Gauss-Jordan Elimination.
Examples:
and / are matrices in reduced row echelon form.is not in reduced row echelon form but in row echelon form since the matrix has the first 3 properties and all the other entries above the leading 1 in the third column are not 0.
is not in row echelon form (also not in reduced row echelon form) since the leading 1 in the second row is not in the left of the leading 1 in the third row and all the other entries above the leading 1 in the third column are not 0.
Example:
Transform matrix A into rref:
Solve the following linear system by finding the rref equivalent of the corresponding matrix.