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SAP Management Information Tutorial

Course: BUS 110, Spring 2010
School: UNC Greensboro
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Information Management Systems UNIT VI Lesson 39 - Tutorial on ERP Packages and Softwares A Short SAP Tutorial What is SAP? SAP is the leading Enterprise Information and Management Package worldwide. Use of this package makes it possible to track and manage, in real-time, sales, production, finance accounting and human resources in an enterprise. What Makes SAP Different? Traditional computer information systems...

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Information Management Systems UNIT VI Lesson 39 - Tutorial on ERP Packages and Softwares A Short SAP Tutorial What is SAP? SAP is the leading Enterprise Information and Management Package worldwide. Use of this package makes it possible to track and manage, in real-time, sales, production, finance accounting and human resources in an enterprise. What Makes SAP Different? Traditional computer information systems used by many businesses today have been developed to accomplish some specific tasks and provide reports and analysis of events that have already taken place. Examples are accounting general ledger systems. Occasionally, some systems operate in a "real-time" mode that is, have up to date information in them and can be used to actually control events. A typical company has many separate systems to manage different processes like production, sales and accounting. Each of these systems has its own databases and seldom passes information to other systems in a timely manner. SAP takes a different approach. There is only one information system in an enterprise, SAP. All applications access common data. Real events in the business initiate transactions. Accounting is done automatically by events in sales and production. Sales can see when products can be delivered. Production schedules are driven by sales. The whole system is designed to be real-time and not historical. SAP structure embodies what are considered the "best business practices". A company implementing SAP adapts it operations to it to achieve its efficiencies and power. The process of adapting procedures to the SAP model involves "Business Process Re-engineering" which is a logical analysis of the events and relationships that exist in an enterprise's operations. SAP Application Modules SAP has several layers. The Basis System is the heart of the data operations and should be not evident to higher level or managerial users. Other customizing and implementation tools exist also. The heart of the system from a manager's viewpoint are the application modules. These modules may not all be implemented in a typical company but they are all related and are listed below : FI Financial Accountingdesigned for automated management and external reporting of general ledger, accounts receivable, accounts payable and other sub-ledger accounts with a user defined chart of accounts. As entries are made relating to sales production and payments journal entries are automatically posted. This connection means that the "books" are designed to reflect the real situation. CO Controllingrepresents the company's flow of cost and revenue . It is a management instrument for organizational decisions. It too is automatically updated as events occur. AM Asset Managementdesigned to manage and supervise individual aspects of fixed assets including purchase and sale of assets, depreciation and investment management. 11.602 Copy Right: Rai University 544 Management Information Systems PS Project Systemis designed to support the planning, control and monitoring of long-term, highly complex projects with defined goals. WF Workflowlinks the integrated SAP application modules with cross-application technologies, tools and services IS Industry Solutionscombine the SAP application modules and additional industry-specific functionality. Special techniques have been developed for industries such as banking, oil and gas, pharmaceuticals, etc. HR Human Resourcesis a complete integrated system for supporting the planning and control of personnel activities. PM Plant MaintenanceIn a complex manufacturing process maintenance means more than sweeping the floors. Equipment must be services and rebuilt. These tasks affect the production plans. MM Materials Managementsupports the procurement and inventory functions occurring in day-to-day business operations such as purchasing, inventory management, reorder point processing, etc. QM Quality Managementis a quality control and information system supporting quality planning, inspection, and control for manufacturing and procurement. PP Production Planningis used to plan and control the manufacturing activities of a company. This module includes; bills of material, routings, work centers, sales and operations planning, master production scheduling, material requirements planning, shop floor control, production orders, product costing, etc. SD Sales and Distributionhelps to optimize all the tasks and activities carried out in sales, delivery and billing. Key elements are; pre-sales support, inquiry processing, quotation processing, sales order processing, delivery processing, billing and sales information system. Each of these Modules may have sub-modules designed for specific tasks as detailed below. System-Wide Features SAP uses certain system wide features that should be understood at the outset. These are used to logically, safely and flexibly organize the data in a business enterprise. Customizing is the configuring of the system to represent your organization's legal structure, reporting requirements and business processes. Internal reporting is a managerial tool in the daily operations. External reporting is required by governmental units controlling the legal structure of the corporation, such as, the IRS state taxing authorities, SEC etc. Organizational Elements o Financial client is a legal and organizationally independent unit at the highest level in SAP company is an independent legal entity within a client business areas are used to produce profit and loss statements and balanced sheets across marketing lines o Materials Management Purchasing units Plants o Sales and Distribution Sales Organization Distribution channel Division 11.602 Copy Right: Rai University 545 Management Information Systems Master Data is records that remain in the database over an extended period of time. Examples: o Customer Master o Vendor Master o Material master o Account Master This structure eliminates redundant data and is shared by all SAP Modules. It is a critical aspect of the robustness of the system. Employee Self Serviceyour employees have access to the own HR records over the Internet. Classification is the assignment of object s to a class. Each class has standard characteristics. Matchcodes are query tools used to find specific information using search methods. Security is administered for objects, profiles and authorizations. Users are only authorized to see or change the parts of the system required by their job responsibilities. Business Processes and SAP Functionality In order to understand a system like SAP a thorough understanding of the events and relationships that take place in a business is required. It is not enough to just realize the Sales, Production, Finance and Accounting have jobs to do in a business. The exact details of each action, the timing of that action and its interrelationships with every other process must be understood. In many large operations there may be no person that has a complete grasp of the situation. Before an operation can be automated or computerized a thorough study of the business must be undertaken. This task is called Business Process Engineering. Sequential Walk Through Sales Pre-sales activityplanning and availability support for the sales personnel Sales OrderThe actual entry of the sales order into the system done by the salesperson at the point of sales perhaps using a PC and Internet connections. o Determining where the most efficient source of the ordered product is in inventory and shipping it. o Delivery o Customer Billing o Customer Payment Production o Sales and Operations Planning SOP where the sales forecasts are used in a production planning model to check feasibility. o Master Production Scheduling MPSThe actual pla n for the whole production process o Material Requirements Planning MRPWhere the production plan is actually converted into raw materials input requirements. o Planned OrderWhen materials are available and capacity exists this plan is created and then converted into a o Production Order. o Shop Floor Control where the actual production takes place and is registered into the system as finished goods. o o 11.602 Copy Right: Rai University 546 Management Information Systems o Purchasing RequisitionOnce the Production manager plans to manufacture something a requisition for the raw materials required but not on hand must be prepared. Vendor Selectionmade by the purchasing department Purchase order sent goods receipt increasing inventory Invoice verification as it is received from vendor Payment to vendor. Finance and Accounting Sales events must be captured at the proper time into the ledger system Inventory must be adjusted to match goods shipped Inventory must be adjusted to match raw materials received Inventory must be adjusted to move value from raw materials to work in process Inventory must be adjusted to increase finished goods when they are produced Accounts Payable must be set up for purchases Accounts Receivable must reflect goods billed but not yet paid for o Business Process Engineering must not only identify all these steps but must also find the most efficient way to minimize redundant actions. For example, when sales are made, inventory and manufacturing plans should be automatically updated. When manufacturing plans are updated raw materials should be automatically ordered from vendors. When finished goods are shipped customers should be automatically billed at the same instant. Real situations are far more complex than the simple explanation above. Quick Tour of the SAP User Interface The SAP R/3 system presents a Windows interface with several of the familiar Windows functions for screen manipulation. The apparent simplicity of the interf ace hides the power of the menus residing within the menubar at the top of the screen. The initial screen shows a menubar with the following selections. The first level sub menus are listed below to give you an idea of where to start: o Office Workplace Telephone Integration Appointment Calendar Room Reservations Start Workflow Business Documents Logistics Materials Management Sales/distribution Logistics Execution Production Production-process Plant Maintenance Customer Service o 11.602 Copy Right: Rai University 547 Management Information Systems Quality Management Logistics controlling Project Management Environment Health & Safety Central Functions Accounting Financial Accounting Treasury Controlling Enterprise Control Investmt Mgt. Project management Real Estate Human Resources Managers Desktop Personnel admin. Time management Payroll Training and Event Management Organizational Management Travel Information system Information Systems Executive Information Systems Logistics Accounting Human Resources Project System Ad Hoc Reports General Report System Tools ABAP/4 Workbench Accelerated SAP Administration ALE Business Communication Business Documents Business Framework Business Workflow CCMS Web Development SAPScript Hypertext o o o o Case : SAP R/3 Implementation at Geneva Pharmaceuticals1 Company Background Geneva Pharmaceuticals, Inc., one of the worlds largest generic drug manufacturers, is the North American hub for the Generics division of Swiss pharmaceutical and life sciences company Novartis International AG. Originally founded by Detroit pharmacist Stanley Tutag in 1946, Geneva moved its 11.602 Copy Right: Rai University 548 Management Information Systems headquarters to Broomfield, Colorado in 1974. The company was subsequently acquired by Ciba Corporation in 1979, which in 1996, merged with Sandoz Ltd. in the largest ever healthcare merger to form Novartis. Alex Krauer, Chairman of Novartis and former Chairman and CEO of Ciba, commented on the strengths of the merger: Strategically, the new company moves into a worldwide leadership position in life sciences. Novartis holds the number two position [globally] in pharmaceuticals, number one in crop protection, and has tremendous development potential in nutrition. The name Novartis comes from the Latin term novae artes or new arts, which eloquently captures the companys corporate vision: to develop new skills in the science of life. Novartis inherited, from its parent companies, a 200-year heritage of serving consumers in three core business segments: healthcare, agribusiness, and nutrition. Today, the Basel (Switzerland) based life sciences company employs 82,500 employees worldwide, runs 275 affiliate operations in 142 countries, and generates annual revenues of 32 billion Swiss Francs. The companys American Depository Receipts trade on the New York Stock Exchange under the ticker symbol NVTSY. Novartis global leadership in branded pharmaceuticals is complemented by its generic drugs division, Novartis Generics. This division is headquartered in Kundl (Austria), and its U.S. operations are managed by Geneva Pharmaceuticals. In 1998, Geneva had revenues of $300 million, employed nearly 1000 employees, and manufactured over 4.6 billion dosage units of generic drugs. Geneva portfolio currently includes over 200 products in over 500 package sizes, covering a wide range of therapeutic categories, such as nervous system disorders, cardio-vascular therapies, and non-steroidal anti-inflammatory drugs. Its major products include ranitidine, atenolol, diclofenac sodium, ercaf, metoprolol tartrate, triamterene with hydrochlorothiazide, and trifluoperazine. Genevas business and product information can be obtained from the company web site at www.genevaRx.com. Generic drugs are pharmaceutically and therapeutically equivalent versions of brand name drugs with established safety and efficacy. For instance, acetaminophen is the equivalent of the registered brand name drug Tylenol, aspirin is equivalent of Ecotrin, and ranitidine HCl is equivalent of Zantac. This equivalence is tested and certified within the U.S. by the Food and Drug Administration (FDA), following successful completion of a bioequivalence study, in which the blood plasma levels of the active generic drug in healthy people are compared with that of the corresponding branded drug. Genevas business strategy has emphasized growth in two ways: (1) focused growth over a select range of product types, and (2) growth via acquisitions. Internal growth was 14 percent in 1998, primarily due to vigorous growth in the penicillin and cephalosporin businesses. In pursuit of further growth, Geneva spend $52 million in 1997 to upgrade its annual manufacturing capacity to its current capacity of 6 billion units, another and $23 million in 1998 in clinical trials and new product development. Industry and Competitive Position The generic drug manufacturing industry is fragmented and highly competitive. In 1998, Geneva was the fifth largest player in this industry, up from its eighth rank in 1997 but still below its second rank in 1996. The companys prime competitors fall into three broad categories: (1) generic drugs divisions of major branded drug companies (e.g., Warrick a division of Schering-Plough and Apothecon a division of Bristol Myers Squibb), (2) independent generic drug manufacturers (e.g., Mylan, Teva Pharmaceuticals, Barr Laboratories, and Watson Pharmaceuticals), and (3) drug distributors vertically integrating into generics manufacturing (e.g., AndRx). The industry also has about 200 smaller players specializing in the manufacture of niche generic products. While Geneva benefited from the financial strength of Novartis, independent companies typically used public stock markets for funding their growth strategies. In 1998, about 45 percent of prescriptions for medications in the U.S. were filled with generics. The trend toward generics can be attributed to the growth of managed care providers such as health maintenance organizations (HMO), who generally prefer lower cost generic drugs to more expensive brand name alternatives (generic drugs typically cost 30-50 less than equivalent brands). However, no single generics manufacturer has benefited from this trend, because distributors and pharmacies view 11.602 Copy Right: Rai University 549 Management Information Systems generic products from different manufacturers as identical substitutes and tend to autosubstitute or freely replace generics from one company with those from another based on product availability and pricing at that time. Once substituted, it is very difficult to regain that customer account because pharmacies are disinclined to change product brand, color, and packaging, to avoid confusion among consumers. In addition, consumer trust toward generics has remained lower, following a generic drug scandal in the early 1990s (of which Geneva was not a part). Margins in the generics sector has therefore remained extremely low, and there is a continuous pressure on Geneva and its competitors to reduce costs of operations. Opportunities for international growth are limited because of two reasons. First, consumers in some countries such as Mexico are generally skeptical about the lack of branding because of their cultural background. Second, U.S. generics manufacturers are often undercut by competitors from India and China, where abundance of low-cost labor and less restrictive regulatory requirements (e.g., FDA approval) makes drug manufacturing even less expensive. Continuous price pressures has resulted in a number of recent industry mergers and acquisitions in the generic drugs sector in recent years, as the acquirers seek economies of scale as a means of reducing costs. The search for higher margins has also led some generics companies to venture into the branded drugs sector, providing clinical trials, research and development, and additional manufacturing capacity for branded drugs on an outsourced basis. Major Business Processes Genevas primary business processes are manufacturing and distribution. The companys manufacturing operations are performed at a 600,000 square foot facility in Broomfield (Colorado), while its two large distribution centers are located in Broomfield and Knoxville (Tennessee). Genevas manufacturing process is scientific, controlled, and highly precise. A long and rigorous FDA approval process is required prior to commercial production of any drug, whereby the exact formulation of the drug or its recipe is documented. Raw materials are sourced from suppliers (sometimes from foreign countries such as China), tested for quality (per FDA requirements), weighed (based on dosage requirements), granulated (i.e., mixed, wetted, dried, milled to specific particle sizes, and blended to assure content uniformity), and compressed into a tablet or poured into a gelatinous capsule. Some products require additional coatings to help in digestion, stabilizing, regulating the release of active ingredients in the human body, or simply to improve taste. Tablets or capsules are then imprinted with the Geneva logo and a product identification number. Following a final inspection, the medications are packaged in childproof bottles with a distinctive Geneva label, or inserted into unit-dose blister packs for shipment. Manufacturing is done in batches, however, the same batch can be split into multiple product types such as tablets and capsules, or tablets of different dosages (e.g., 50 mg and 100 mg). Likewise, finished goods from a batch can be packaged in different types of bottles, based on customer needs. These variations add several layers of complexity to the standard manufacturing process and requires tracking of three types of inventory: raw materials, bulk materials, and finished goods, where bulk materials represent the intermediate stage prior to packaging. In some cases, additional intermediates such as coating solution is also tracked. Master production scheduling is focused on the manufacture of bulk materials, based on forecasted demand and replenishment of safety stocks at the two distribution centers. Finished goods production depends on the schedule-to-performance, plus availability of packaging materials (bottles and blister packs), which are sourced from outside vendors. Bulk materials and finished goods are warehoused in Broomfield and Knoxville distribution centers (DC) prior to shipping. Since all manufacturing is done was done at Broomfield, inventory replenishment of manufactured products is done first at Broomfield and then at Knoxville. To meet additional customer demand, Geneva also purchases finished goods from smaller manufacturers, who manufacture and package generic drugs under Genevas level. Since most of these outsourcers are located along the east coast, and hence, they are distributed first to the Knoxville and then to Broomfield. Purchasing is simpler 11.602 Copy Right: Rai University 550 Management Information Systems than manufacturing because it requires no bill of materials, no bulk materials management, and no master scheduling; Geneva simply converts planned orders to purchase requisitions, and then to purchase orders, that are invoiced upon delivery. Geneva supplies to a total of about 250 customers, including distributors (e.g., McKesson, Cardinal, Bergen), drugstore chains (e.g., Walgreen, Rite-Aid), grocery chains with in-store pharmacies (e.g., Safeway, Kroger), mail order pharmacies (e.g., Medco, Walgreen), HMOs (e.g., Pacificare, Cigna), hospitals (e.g., Columbia, St. Lukes), independent retail pharmacies, and governmental agencies (e.g., U.S. Army, Veterans Administration, Federal prisons). About 70 percent of Genevas sales goes to distributors, another 20 percent goes to drugstore chains, while HMOs, government, retail pharmacies, and others account for the remaining 10 percent. Distributors purchase generic drugs wholesale from Geneva, and then resell them to retail and mail order pharmacies, who are sometimes direct customers of Geneva. The volume and dollar amount of transaction vary greatly from one customer to another, and while distributors are sometimes allow Geneva some lead time to fulfill in a large order, retail pharmacies typically are unwilling to make that concession. One emerging potential customer segment is Internetbased drug retailers such as Drugstore.com and PlanetRx.com. These online drugstores do not maintain any inventory of their own, but instead accept customer orders and pass on those orders to any wholesaler or manufacturer that can fill those orders in short notice. These small, customized, and unpredictable orders do not fit well with Genevas wholesale, high-volume production strategy, and hence, the company has decided against direct retailing to consumers via mail order or the Internet, at least for the near future. As is standard in the generics industry, Geneva uses a complex incentive system consisting of rebates and chargebacks to entice distributors and pharmacies to buy its products. Each drug is assigned a published industry price by industry associations, but Geneva rebates that price to distributors on their sales contracts. For instance, if the published price is $10, and the rebates assigned to a distributor is $3, then the contract price on that drug is $7. Rebate amounts are determined by the sales management based on negotiations with customers. Often times, customers get proposals to buy the product cheaper from a different manufacturer and ask Geneva for a corresponding discount. Depending on how badly Geneva wants that particular customer or push that product, it may offer a rebate or increase an existing rebate. Rebates can vary from one product to another (for the same customer) and/or from one order volume to another (for the same product). Likewise, pharmacies ordering Genevas products are paid back a fraction of the sales proceeds as chargebacks. The majority of Genevas orders come through EDI. These orders are passed though multiple filters in an automated order processing system to check if the customer has an active customer number and sufficient credit, if the item ordered is correct and available in inventory. Customers are then assigned to either the Broomfield or Knoxville DC based on quantity ordered, delivery expiration dates, and whether the customer would accept split lots. If the quantity ordered is not available at the primary DC (say, Knoxville), a second allocation is made to the secondary DC (Broomfield, in this case). If the order cannot be filled immediately, a backorder will be generated and the Broomfield manufacturing unit informed of the same. Once filled, the distribution unit will print the order and ship it to the customer, and send order information to accounts receivable for invoicing. The overall effectiveness of the fulfillment process is measured by two customer service metrics: (1) the ratio between the number of lines on the order that can be filled immediately (partial fills allowed) to the total number of lines ordered by the customer (called firstfill), and (2) the percentage of items send from the primary DC. Fill patterns are important because customers typically prefer to get all items ordered in one shipment. Matching customer demand to production schedules is often difficult because of speculative buying on the part of customers. Prices of drugs are typically reassessed at the start of every fiscal year, and a distributor may place a very large order at the end of the previous year to escape a potential price increase at the start of the next year (these products would then be stockpiled for reselling at higher prices next year). Likewise, a distributor may place a large order at the end of its financial year to transfer cash-on-hand to cost-of-goods-sold, for tax purposes or to ward off a potential acquisition threat. Unfortunately, most generics companies do not have the built-in capacity to deliver such orders within short time frames, yet inability to fulfill orders 11.602 Copy Right: Rai University 551 Management Information Systems may lead to the loss of an important customer. Safety stocks help meet some of these unforeseen demands, however maintaining such inventory consumes operating resources and reduce margins further. SAP R/3 Implementation Up until 1996, Genevas information systems (IS) consisted of a wide array of software programs for running procurement, manufacturing, accounting, sales, and other mission-critical processes. The primary hardware platform was IBM AS/400, running multiple operational databases (mostly DB/2) and connected to desktop microcomputers via a token-ring local area network (LAN). Each business unit had deployed applications in an ad hoc manner to meet its immediate needs, which were incompatible across business units. For instance, the manufacturing unit (e.g., materials requirements planning) utilized a manufacturing application called MacPac, financial accounting used Software/2000, and planning/budgeting used FYI-Planner. These systems were not interoperable, and data that were shared across systems (e.g., accounts receivable data was used by order management and financial accounting packages, customer demand was used in both sales and manufacturing systems) had to be double-booked and rekeyed manually. This led to higher incidence of data entry errors, higher costs of error processing, and greater data inconsistency. Further, data was locked within functional silos and were unable to support processes that cut across multiple business units (e.g., end-to-end supply chain management). It was apparent that a common, integrated company-wide solution would not only improve data consistency and accuracy, but also reduce system maintenance costs (e.g., data reentry and error correction) and enable implementation of new value-added processes across business units. In view of these limitations, in 1996, corporate management at Geneva initiated a search for technology solutions that could streamline its internal processes, lower costs of operations, and strategically position the company to take advantage of new value-added processes. More specifically, it wanted an enterprise resource planning (ERP) software that could: (1) implement best practices in business processes, (2) provide operational efficiency by integrating data across business units, (3) reduce errors due to incorrect keying or rekeying of data, (4) reduce system maintenance costs by standardizing business data, (5) be flexible enough to integrate with new systems (as more companies are acquired), (6) support growth in product and customer categories, and (7) is Y2K (year2000) compliant. The worldwide divisions of Novartis were considering two ERP packages at that time: BPCS from Software Systems Associates and R/3 system from SAP. Eventually, branded drug divisions decided to standardize their data processing environment using BPCS, and generics agreed on deploying R/3.2 A brief description of the R/3 software is provided in the below exhibits. 11.602 Copy Right: Rai University 552 Management Information Systems 11.602 Copy Right: Rai University 553 Management Information Systems 11.602 Copy Right: Rai University 554 Management Information Systems 11.602 Copy Right: Rai University 555 Management Information Systems 11.602 Copy Right: Rai University 556
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Waterloo - MATH - 235/237
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Waterloo - MATH - 235/237
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Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 6 Solutions1. Show that the following are equivalent for a symmetric matrix A: (1) A is orthogonal (2) A2 = I (3) All the eigenvalues of A are 1 Solution: (1) (2) (2) (3) If A is orthogonal then I = AAT = AA, since A is symmetric. Av
Waterloo - MATH - 235/237
Math 235Assignment 7Due: Wednesday, June 30th1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Cla
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Math 235Assignment 7 Solutions1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Classify each quadr
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 8Due: Wednesday, July 14th1. Sketch the graph of 9x2 + 4xy + 6y 2 = 21 showing both the original and new axes. Solution: The corresponding symmetric matrix is polynomial is C ( ) = 9 2 = 2 15 + 50 = ( 10)( 5). 2 6 A 1 I = 1 2 1 2 . 2
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Math 235Assignment 9Due: Wednesday, July 21st1. Suppose that a real 2 2 matrix A has 2 + i as an eigenvalue with a corresponding 1+i eigenvector . Determine A. i 0 2 1 2. Determine a real canonical form of A = 2 2 1 and give a change of basis matrix 0
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Math 235Assignment 9 Solutions1. Suppose that a real 2 2 matrix A has 2 + i as an eigenvalue with a corresponding 1+i eigenvector . Determine A. i Solution: Since A is real, we know that A has real canonical form B = brought into this form by P = 11 . W
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Math 235Assignment 10 Not To Be Submitted 1+i 1i 1. Consider C3 with its standard inner product. Let z = 2 i , w = 2 3i. 1 + i 1 a) Evaluate z , w and w, 2iz . b) Find a vector in spancfw_z, w that is orthogonal to z . c) Write the formula for the proj
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Math 235Assignment 10 Solutions 1+i 1i 1. Consider C3 with its standard inner product. Let z = 2 i , w = 2 3i. 1 + i 1 a) Evaluate z , w and w, 2iz . Solution: We have z , w = (1 + i)(1 + i) + (2 i)(2 + 3i) + (1 + i)(1) = 2i 1 + 8i + 1 i = 9i w, 2iz =
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Math 235 - Final Exam Fall 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not ask
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Math 235Final F09 AnswersNOTE: These are only answers to the problems and not full solutions! On the nal exam you will be expected to show all steps used to obtain your answer. 1. Short Answer Problems 3 i i a) A = . 2 1 b) A is Hermitian since A = A, a
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Math 235 - Final Exam Spring 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not a
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Math 235Final S09 AnswersNOTE: These are only answers to the problems and not full solutions! On the nal exam you will be expected to show all steps used to obtain your answer. 1. a) A basis for the nullspace is cfw_x, hence the nullity of L is 1. Thus,
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Math 235Final Exam InformationThursday August 5, 9:00 AM - 11:30 AMLOCATION: PAC 1, 2, 3Material Covered: Entire Course, with an emphasis on material after term test 2. Information: - Surfaces in R3 are not covered. - Fourier Series are not covered. -
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Math 235 1. Short Answer ProblemsSample Term Test 1 - 1a) Give the denition of an inner product , on a vector space V . b) Let B = cfw_v1 , . . . , vn be orthonormal in an inner product space V and let v V such that v = a1 v1 + + an vn . Prove that ai
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Math 235Sample Term Test 1 - 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the denition of an inner product , on a vect
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Math 235 1. Short Answer ProblemsSample Term Test 1 - 2 1 0 0 1 a) Write a basis for the rowspace, columnspace and nullspace of A = 0 0 1 1 . 000 0 b) Let B = cfw_v1 , . . . , vn be orthonormal in an inner product space V and let v = a1 v1 + + an vn .
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Math 235Sample Term Test 1 - 2 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems 1 0 0 1 a) Let A = 0 0 1 1 . Write a basis for the R
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Math 235 1. Short Answer ProblemsSample Term Test 2 - 1a) Let S be a subspace of an inner product space V . What is the denition of S . b) State the Principal Axis Theorem. c) Determine the matrix for the quadratic form Q(x, y, z ) = 3x2 y 2 + z 2 2xy +
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Math 235Sample Term Test 2 - 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let S be a subspace of an inner product space V .
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Math 235 1. Short Answer ProblemsSample Term Test 2 - 2a) State the Principal Axis Theorem. b) Let A be an m n matrix. Prove that AT A is symmetric. c) State the denition of a quadratic form Q(x) on Rn being negative denite. d) Consider the quadratic fo
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Math 235Sample Term Test 2 - 2 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) State the Principal Axis Theorem. Solution: A mat
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Math 235Midterm InformationTuesday, June 8th, 4:30 - 6:20 p.mMaterial Covered: Sections 4-5, 4-6, 4-7, 7-4 (not including Fourier series), 7-1. You need to know: - All denitions and statements of theorems. - How to nd a basis of the rowspace, column sp
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Math 235Term Test 2 InformationTuesday, July 6th, 4:30 - 6:20 p.mRoom Assignments: MC 4059: A - G MC 4061: H - Lin MC 4045: Liu - P MC 4020: Q - Wang MC 4021 Wardell - Z Material Covered: Sections 7-2, 7-3, Triangularization, 8-1, 8-2. You need to know
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Math 235Tutorial: Term Test 1 Review1: State the denition of: a) One-to-one b) Onto c) An orthogonal matrix (what are 2 other equivalent denitions?) d) An inner product 1 0 1/2 1/2 1/2 1/2 , , 2 1 1/2 1/2 1/2 1/2 0 2 T under the inner product A, B = tr
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SOSMATH235MIDTERM2REVIEWPACKAGE HelloMATH235students,mynameisTaiCaiandIamtheSOStutorthistermforMATH235.This packageisdesignedtosupplementyourstudyingforthesecondmidtermonNovember16,2010. Wheneverpossible,Ihaveincludedexamplesthatarenotfromclassoryourtextb
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Math 235 1. Short Answer ProblemsTerm Test 1 Solutions[1] a) State the denition of the rank of a linear mapping L : V W . Solution: rank(L) = dim Range(L).[2] b) Let B = cfw_v1 , . . . , vn be a basis for a vector space V and let L : V W be an isomorp
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Math 235 31 5 2 1. Let A = 2 1 32 4 7 1 5 2 3 3 2Assignment 1Due: Wednesday, Sept 22nd 0 1 0 0 1 1 0 0 01 0 2 . 1 1 00 3 1 0 4 , then the RREF of A is R = 0 7 1 0a) Find rank(A) and dim(Null(A). b) Find a basis for Row(A). c) Find a basis for Null(A).
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Math 235 31 5 2 1. Let A = 2 1 32 4 7 1 5 2 3 3 2Assignment 1 Solutions 3 1 0 4 , then the RREF of A is R = 0 7 1 0 0 1 0 0 1 1 0 0 01 0 2 . 1 1 00a) Find rank(A) and dim(Null(A). Solution: rank(A) = 3 and dim(Null(A) = 5 3 = 2 b) Find a basis for Row(A
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 2 Solutions1. For each of the following linear transformations, determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . a) The projection proj(3,2) : R2 R2 onto the line x = t Solut
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Math 235Assignment 3Due: Wednesday, Oct 6th1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) M (2, 2) and P3 . b) The vector space P
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Math 235Assignment 3 Solutions1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) M (2, 2) and P3 . Solution: We dene L : M (2, 2) P3 by
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Math 235Assignment 4Due: Wednesday, Oct 13th1. Prove that the product of two orthogonal matrices is an orthogonal matrix. 2. Observe that the dot product of two vectors x, y Rn can be written as x y = xT y. Use this fact to prove that if an n n matrix
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Math 235Assignment 4 Solutions1. Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let P and Q be orthogonal matrices. Then we have (P Q)T (P Q) = QT P T P Q = QT Q = I, since P T P = I and QT Q = I . Thus P Q is also
Waterloo - MATH - 235/237