Math 221: Matrix Algebra Midterm 1 - May 23, 2012
Instructions: There are ﬁve questions and 100 points on this exam. You will need only a pen or pencil and eraser; nothing else is permitted. Unless otherwise indicated, write your ﬁnal answers clearly in complete sentences; failure to do so will cost points. Point values indicated for each question are estimates and subject to change. (1) (12 points) Suppose that T : Rn → Rn which is not one-to-one, and let A denote the standard matrix of T . Indicate whether the following are true or false by writing the complete word True or False (you will lose points for simply writing T or F). (a) A is a square matrix. (b) The columns of A are linearly dependent. (c) A has a pivot in every column. (d) T is not onto.

(2) (16 points) For each of the following mappings, write linear if the mapping is a linear transformation, and otherwise write not linear. You do not need to justify your answers. (a) T : R2 → R3 deﬁned by T x y x+y = x2 0

(b) T : R2 → R3 deﬁned by T x y x+y = x+1 0

(c) T : R → R3 deﬁned by

x T (x) = −2x 0

(d) T : Rn → Rm deﬁned by T (x) = 0.

(3) (24 points) Suppose that a linear system has a coeﬃcient matrix A whose reduced echelon form REF (A) is 1 −1 0 −1 0 0 0 1 −1 0 REF (A) = 0 0 0 0 1 0 0 0 0 0

(a) Express the solution set for the homogeneous linear system Ax = 0 in parametric vector form.

(b) Suppose that a1 , a2 , a3 , a4 , and a5 are the columns of A, so that A= a1 a2 a3 a4 a5 Express the zero vector 0 ∈ R4 as a linear combination of the columns of A in which not all the coeﬃcients are zero.

(c) Do the columns of A span R4 ? Justify your answer.

(4) (24 points) Suppose that T : R2 → R3 is a linear transformation such that T (e1 − 2e2 ) = 3e1 − e3 Let A be the standard matrix of T . (a) How many rows and columns does A have? T (−e1 + e2 ) = −2e2

...Formulas
Read the following instructions in order to complete this discussion, and review the example of how to complete the math required for this assignment:
• Read about Cowling’s Rule for child sized doses of medication (number 92 on page 119 of Elementary and Intermediate Algebra).
• Solve parts (a) and (b) of the problem using the following details indicated for the first letter of your last name:
|If your last |For part (a) of problem 92 use this information to calculate the child’s |For part (b) of problem 92 use this |
|name starts with letter|dose. |information to calculate the child’s |
| | |age. |
|A or Z |adult dose 400mg ibuprofen; 5 year old child |800mg adult, 233mg child |
|C or X |adult dose 500mg amoxicillin; 11 year old child |250mg adult, 52mg child |
|E or V |adult dose 1000mg acetaminophen; 8 year old child |600mg adult, 250mg child |
|G or T |adult dose 75mg Tamiflu; 6 year old child |500mg adult, 187mg child |
|I or R |adult dose 400mg ibuprofen; 7 year old child...

...WEST VISAYAS STATE UNIVERSITY
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EXTENSION CAMPUS HIMAMAYLAN CITY
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HIMAMAYLAN CITY, NEGROS OCCIDENTAL
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MidtermExamination
100
Society and Culture with Family Planning
NAME: ________________________________ SCORE:
I. Who Am I?
1. I am famous of my study about suicides. And I am also known for my concepts of the “collective representations” as the social power of ideas stemming from their developments through interactions of many minds. Who Am I? ___________________________.
2. I believe that a stable society may undergo a series of conflict before they reach stability. And I also believe in the idea of communism. Who Am I? ____________________________.
3. I believe with the idea of looking- glass self constitutes the imagination of how one appears to others and the imagination of the judgement of that appearance.
Who Am I? _______________________________.
4. I am concerned with social change and the plight of women and children in English factories during the early phases of industrialization. I am the first female sociologist.
Who Am I? _______________________________.
5. I believed that sociologist could never capture the reality of society but should focus on ideal types that best capture the...

...
Working with algebra you must understand why the properties of real numbers are so important. I will demonstrate my solutions to three problems and I will include my mathematical work. Also I will use five vocabulary words that help me find solutions to the problems. Properties of real numbers are useful for simplifying algebraic expression because a lot of thing we do in life are equations. We use a lot of mathematical terms in the real world. Lastly I will show every step I took to simplify and identify each property of real numbers.
The properties of algebra are important to know and understand. “Algebra is useful because it can be used to solve problems. Since problems are often communicated verbally, we must be able to translate verbal expressions into algebraic expressions and translate algebraic expressions into verbal expressions.” (Dugopolski, 2012, Chapter 1.6, ) Each expression has properties that must be simplified and solving methods. You have to know how to follow the order of operation and simplify the equations, variables and like terms in order to complete the mathematical work. Simplify is very important in all expression, must be in simplest form when completely an equations. Also you have to move and combine like terms. The coefficient of a number is in front of a variable. The coefficient is a factor that can produce a result. The distributive property is a step that multiples a term to be...

...Deterministic techniques assume that no uncertain exists in model parameters.
A: True
An inspector correctly identifies 90% of the time. For the next 10 products, the probability that he makes fewer than 2 incorrect inspections is .736.
A: Use Binomial table to discover , add 3 probabilities for 0,1,2
A continuous random variable may assume only integer values within a given interval.
A: False
A decision tree is a diagram consisting of circles decision nodes, square probability nodes and branches.
A: False
A table of random numbers must be normally distributed and efficiently generated
A: False
Simulation results will always equal analytical results if 30 trials of the simulation have been conducted.
A: False
Data cannot exhibit both trend and cyclical patterns.
A: False
The Delphi method develops a consensus forecast about what will occur in the future.
A: True
A company markets education software products and is ready to place three new products on the market, past experience has shown that for this particular software. The chance of success is 80%. Assume that the probability of succeed is independent for each product, what is the probability that exactly 1 of the 3 products is successful.
A: binomial answer use table
_ is a measure of dispersion of random variable values about the expected value.
A: Standard Deviation
The _ is the expected value of the regret for each decision.
A: expected opportunity loss
A seed value is a
A: number used...

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ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which resists relative lateral motion of two solid surfaces in contact and is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces; lubricated friction which is a case of fluid friction where a fluid separates two solid surfaces; skin friction which is a component of drag, the force resisting the motion of a fluid across the surface of a body; internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation and sliding friction.
When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into heat. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to heat whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear,...

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The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.
A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations. An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted. It has to be clear and certain in terms. It must also be communicated to the offeree before it is being accepted. In addition, the acceptance has to be unqualified, unconditional and made by a positive act. In the case of Beauty and Stylish, a positive act refers to the signing of the contract. All terms of the offer must be accepted without any changes and cannot be subjected to any condition, taking effect only upon fulfillment of that condition. When Beauty and Stylish enter into the agreement, they must intend to bind and bound legally to each other by their agreement. This is the intention to create legal relations between two parties. In the meanwhile, this contract must possess consideration. A contract must therefore be a two-sided affair, with each side providing or promising to provide something of value in exchange for what the other is to provide.
Every contract, whether oral or written, contain terms. The terms of a contract set out the rights and duties of the parties. Terms are the promises and undertakings given by each...

...OF FINDING INVERSE PROPORTION.
In linear algebra, an n-by-n (square) matrix A is called invertible or nonsingular or nondegenerate if there exists an n-by-n matrix B such that
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A−1. It follows from the theory of matrices that if
for square matrices A and B, then also
Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = I. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = I.
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any commutative ring.
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular.
Matrix inversion is the process of finding the...

...Math221
Week 2 Lab
Part 1. Random Sampling
1. To perform random sampling, I used the Data Analysis function under the Data tab. Once I clicked on Data Analysis I chose the Sampling function. I then chose the Input Range of data to be random sampled. I checked the Random Sampling Method and input the number of random samples to be selected, 20. I then chose Output Range in the Output Options area of the Sampling box and selected an area on my Excel document to display the output. I clicked OK and the random samples were automatically chosen and displayed in the appropriate chosen location on my Excel document. The Sampling method in the Data Analysis add-on pack is very important to use when truly random samples need to be taken from a group of data for analysis.
2.
Part 2. Cereal and Fiber Type
Fiber Type | Frequency | % fiber |
High Fiber | 22 | 26% |
Medium Fiber | 29 | 35% |
Low Fiber | 33 | 39% |
Total | 84 | 100% |
Part 3. Milk Production
1. The sample mean of the data is 2270.54 lbs. of milk production.
2. The sample standard deviation of the milk production is 653.1822.
3.
Milk Production |
Class Limits | | | Cumulative |
Lower | Upper | Midpoint | Frequency | Frequency |
1147 | 1646 | 1396.5 | 7 | 7 |
1647 | 2146 | 1896.5 | 15 | 22 |
2147 | 2646 | 2396.5 | 13 | 35 |
2647 | 3146 | 2896.5 | 11 | 46 |
3147 |...