| How to Reduce a Matrix to Row Echelon Form Posted: 05 May 2016 05:00 PM PDT The row-echelon form of a matrix is highly useful for many applications. For example, it can be used to geometrically interpret different vectors, solve systems of linear equations, and find out properties such as the determinant of the matrix. - Understand what row-echelon form is. The row-echelon form is where the leading (first non-zero) entry of each row has only zeroes below it. These leading entries are called pivots, and an analysis of the relation between the pivots and their locations in a matrix can tell much about the matrix itself. An example of a matrix in row-echelon form is below.
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- Understand how to perform elementary row operations. There are three row operations that one can do to a matrix.
- Row swapping.
- Scalar multiplication. Any row can be replaced by a non-zero scalar multiple of that row.
- Row addition. A row can be replaced by itself plus a multiple of another row.
- Begin by writing out the matrix to be reduced to row-echelon form.
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- Identify the first pivot of the matrix. The pivots are essential to understanding the row reduction process. When reducing a matrix to row-echelon form, the entries below the pivots of the matrix are all 0.
- For our matrix, the first pivot is simply the top left entry. In general, this will be the case, unless the top left entry is 0. If this is the case, swap rows until the top left entry is non-zero.
- By their nature, there can only be one pivot per column and per row. When we selected the top left entry as our first pivot, none of the other entries in the pivot's column or row can become pivots.
- Perform row operations on the matrix to obtain 0's below the first pivot.
- For our matrix, we want to obtain 0's for the entries below the first pivot. Replace the second row with itself minus the first row. Replace the third row with itself minus three times the first row. These row reductions can be succinctly written as and
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- Identify the second pivot of the matrix. The second pivot can either be the middle or the middle bottom entry, but it cannot be the middle top entry, because that row already contains a pivot. We will choose the middle entry as the second pivot, although the middle bottom works just as well.
- Perform row operations on the matrix to obtain 0's below the second pivot.
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- This matrix is in row-echelon form now.
- In general, keep identifying your pivots. Row-reduce so that the entries below the pivots are 0.
- Reducing a matrix to row-echelon form works with any size matrix, both square and rectangular.
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| How to Write a Research Essay Posted: 05 May 2016 09:00 AM PDT Research essays are extremely common assignments in high school, college, and graduate school, and are not unheard of in middle school. If you are a student, chances are you will sooner or later be faced with the task of researching a topic and writing a paper about it. Knowing how to efficiently and successfully do simple research, synthesize information, and clearly present it in essay form will save you many hours and a lot of frustration. EditResearching a Topic - Choose a topic. If you are lucky enough to be able to choose your research topic, pick one that interests you. The last thing you want is to end up spending hours learning and writing about a subject in which you have zero interest. You will find the whole process much more enjoyable if you actually care about what you are doing.[1]
- Be sure to stay within the guidelines you are given by your teacher or professor. For example, if you are free to choose a topic but the general theme must fall under human biology, do not write your essay on plant photosynthesis.
- Stick with topics that are not overly complicated, especially if the subject is not something you plan to continue studying. There's no need to make things harder on yourself!
- Locate resources. Before you begin writing, read up on the subject of your essay. It is best to gather all your sources and review them before attempting to begin your paper so that you can have things clearly planned out when you actually start writing. You can find information on your topic of choice through a number of different types of resources. Some examples are discussed below.
- Specialty books; these can be found at your local public or school library. A book published on your topic is a great resource and will likely be one of your most reliable options for finding quality information. They also contain lists of references where you can look for more information.
- Academic journals; these are periodicals devoted to scholarly research on a specific field of study. Articles in academic journals are written by experts in that field and scrutinized by other professionals to ensure their accuracy. These are great options if you need to find detailed, sophisticated information on your topic; avoid these if you are only writing a general overview.
- Online encyclopedias; the most reliable information on the internet can be found in online encyclopedias like Encyclopedia.com and Britannica.com. While online wikis can be very helpful, they sometimes contain unverified information that you should probably not rely upon as your primary resources.
- Expert interviews; if possible, interview an expert in the subject of your research. Experts can be professionals working in the field you are studying, professors with advanced degrees in the subject of interest, etc.
- Take notes. When you are researching your topic, take notes on the information you find. It can be difficult to recall all you have read on the subject when it comes time to start writing, especially if you are unfamiliar with the topic; having notes will significantly help with this. Be sure to note where you found the information as you take notes so you can easily cite it as you write.
- Organize your notes by sub-topic to keep them orderly and so you can easily find references when you are writing.
- If you are using books or physical copies of magazines or journals, use sticky tabs to mark pages or paragraphs where you found useful information. You might even want to number these tabs to correspond with numbers on your note sheet for easy reference.
- By keeping your notes brief and simple, you can make them easier to understand and reference while writing. Don't make your notes so long and detailed that they essentially copy what's already written in your sources, as this won't be helpful to you.
- Develop an objective. Depending on the purpose of your research paper, you may find yourself needing to adopt a position or draw some conclusions about your topic. As you research the subject, ask yourself how the information you encounter fits in with the objective of your paper. For example, if you need to present two sides of an argument and then side with one, identify information that corresponds to the different viewpoints surrounding the topic (and organize the sources accordingly in your notes).
- Sometimes the objective of your research will be obvious to you before you even begin researching the topic; other times, you may have to do a bit of reading before you can determine the direction you want your essay to take.
- If you have an objective in mind from the start, you can incorporate this into online searches about your topic in order to find the most relevant resources. For example, if your objective is to outline the environmental hazards of hydraulic fracturing practices, search for that exact phrase rather than just "hydraulic fracturing."
- Talk to your teacher. If you are researching for a class, ask your teacher or professor for advice or suggestions as to the direction you should take with your essay. He or she might be able to help you out by narrowing (or broadening) your focus or by pointing you toward useful resources. This way, you will also be able to gauge whether your teacher approves of the topic you have in mind.
- Avoid asking your teacher to give you a topic. Unless your topic was assigned to you in the first place, part of the assignment is for you to choose a topic relevant to the broader theme of the class or unit. By asking your teacher to do this for you, you risk admitting laziness or incompetence.
- If you have a few topics in mind but are not sure how to develop objectives for some of them, your teacher can help with this. Plan to discuss your options with your teacher and come to a decision yourself rather than having him or her choose the topic for you from several options.
EditOrganizing your Essay - Break up your essay into sub-topics. You will probably need to address several distinct aspects of your research topic in your essay. This is an important tactic for producing a well-organized research essay because it avoids 'stream of consciousness' writing, which typically lacks order.
- Consider what background information is necessary to contextualize your research topic. What questions might the reader have right out of the gate? How do you want the reader to think about the topic? Answering these kinds of questions can help you figure out how to set up your argument.
- Match your paper sections to the objective(s) of your writing. For example, if you are trying to present two sides of a debate, create a section for each and then divide them up according to the aspects of each argument you want to address.
- Create an outline. One of the most helpful things you can do when writing a research paper is to outline the various sections and primary points of the essay. Do this before you begin writing so you can visualize how each of the essay's parts will fit together. This will also allow you to rearrange components of the paper to make it flow logically.[2]
- An outline can be as detailed or general as you want, so long as it helps you figure out how to construct the essay. Some people like to include a few sentences under each heading in their outline to create a sort of "mini-essay" before they begin writing. Others find that a simple ordered list of topics is sufficient. Do whatever works best for you.
- If you have time, write your outline a day or two before you start writing and come back to it several times. This will give you an opportunity to think about how the pieces of your essay will best fit together. Rearrange things in your outline as many times as you want until you have a structure you are happy with.
- Choose a format. Research papers, unlike creative writing pieces, usually adhere to a specific style guides governing the way sources must be cited and various other aspects of writing mechanics. If you are writing a research essay for a class, your teacher will probably specify which style the essay must conform to. Common examples include MLA (Modern Language Association) and APA (American Psychological Association) style formats, but specific fields also have their own style guides. Consult your teacher if you aren't sure what style to use.[3] [4]
- Style guides tell you exactly how to quote passages, cite references, construct works cited sections, etc. If you are assigned a specific format, you must take care to adhere to guidelines for text formatting and citations.
- Some computer programs (such as EndNote) allow you to construct a library of resources which you can then set to a specific format type; then you can automatically insert in-text citations from your library and populate a references section at the end of the document. This is an easy way to make sure your citations match your assigned style format.
- Make a plan. You should set realistic writing goals for yourself so you can stay on task without feeling overburdened. It is a good idea to create a schedule and set aside blocks of time each day to work on specific parts of your essay. This way you aren't stuck writing nonstop for two days to meet your deadline and you can check things off your list as you complete them.
- You may wish to start by simply assigning yourself a certain number of pages per day. Divide the number of pages you are required to write by the number of days you have to finish the essay; this is the number of pages (minimum) that you must complete each day in order to pace yourself evenly.
- If possible, leave a buffer of at least one day between finishing your paper and the due date. This will allow you to review your finished product and edit it for errors. This will also help in case something comes up that slows your writing progress.
EditWriting your Essay - Create an introduction. In this section, introduce your topic and establish the purpose for your essay. If you intend to investigate a debated topic, state this in your introduction. You want the reader to have a good idea of what the essay is about and how it is constructed by reading your introduction. Save your opinions and any conclusions you've drawn for the rest of the essay.[5]
- Keep your introduction relatively short. For most papers, one or two paragraphs will suffice. For really long essays, you may need to expand this.
- Don't assume your reader already knows the basics of the topic unless it truly is a matter of common knowledge. For example, you probably don't need to explain in your introduction what biology is, but you should define less general terms such as "eukaryote" or "polypeptide chain."
- Build the body of your essay. This is the meat of your paper, on which you should place the majority of your focus. The length and detail of your essay will determine the form of its body, but at a minimum this should include any key arguments, any research methods used and results obtained (in cases where you performed original research), and your main research findings.[6] [7]
- You may need to include a special section at the beginning of the essay body for background information on your topic. Alternatively, you can consider moving this to the introductory section, but only if your essay is short and only minimal background discussion is needed.
- This is the part of your paper where organization and structure are most important. Arrange sections within the body so that they flow logically and the reader is introduced to ideas and sub-topics before they are discussed further.
- Depending upon the length and detail of your paper, the end of the body might contain a discussion of findings. This kind of section serves to wrap up your main findings but does not explicitly state your conclusions (which should come in the final section of the essay).
- Avoid repetition in the essay body. Keep your writing concise, yet with sufficient detail to address your objective(s) or research question(s).
- Cite your references properly. One of the biggest mistakes you can make when writing a research paper is to fail to properly cite your sources. Passing off someone else's ideas as your own, whether intentional or not, is plagiarism, and it could land you a failing grade or even expulsion from your school. Take the time to ensure you are citing information the right way by following these guidelines:
- Always use quotation marks when using exact quotes from another source. If someone already said or wrote the words you are using, you must quote them this way! Place your in-text citation at the end of the quote.
- To include someone else's ideas in your essay without directly quoting them, you can restate the information in your own words; this is called paraphrasing. Although this does not require quotation marks, it should still be accompanied by an in-text citation.
- State your conclusions. This section stands apart from the essay body in that it is devoted solely to stating the conclusions you have drawn from your research. Avoid discussing details of your research or presenting results in this section. You may wish to rephrase your study objective and state how your findings address that goal.[8]
- Except for very long essays, keep your conclusion short and to the point. You should aim for one or two paragraphs, if possible.
- Conclusions should directly correspond to research discussed in the essay body. In other words, make sure your conclusions logically connect to the rest of your essay and provide explanations when necessary.
- If your topic is complex and involves lots of details, you should consider including a brief summary of the main points of your research in your conclusion.
- Revisit your thesis or objective. Once you've completed your first draft, you should go back to your introductory paragraph(s) and evaluate whether your essay accomplishes the stated goals you presented in the beginning. A good essay will thoroughly address any questions or unknowns posed in its introduction. If your conclusions do not logically follow the stated purpose or objective of your essay, then you will need to fix this.
- Making changes to the discussion and conclusion sections instead of the introduction often requires a less extensive rewrite. Doing this also prevents you from removing anything from the beginning of your essay that could accidentally make subsequent portions of your writing seem out of place.
- It is okay to revise your thesis once you've finished the first draft of your essay! People's views often change once they've done research on a topic. Just make sure you don't end up straying too far from your assigned topic if you do this.
- You don't necessarily need to wait until you've finished your entire draft to do this step. In fact, it is a good idea to revisit your thesis regularly as you write. This can save you a lot of time in the end by helping you keep your essay content on track.
- Construct a "works cited" section. This is a critical element of any research paper, because this is where you give credit to all the sources from which you borrowed information to write your essay. This is not something that should be left for the end of your writing; rather, you should build your works cited section as you write, adding citations as you reference them in your writing.
- Computer software such as EndNote is available for making citation organization as easy and quick as possible. You can create a reference library and link it to your document, adding in-text citations as you write; the program creates a formatted works cited section at the end of your document.
- Be aware of the formatting requirements of your chosen style guide for works cited sections and in-text citations. Reference library programs like EndNote have hundreds of pre-loaded formats to choose from.
- Put finishing touches on your essay. After you have written your essay, there are some final things to take care of to turn your paper into a polished piece of work that your teacher will appreciate. While not all of these have to wait until the essay is fully written, it is a good idea to take care of the important stuff first -- which is the writing, of course! Final tasks include:
- Create a catchy title. Waiting until you have finished your essay before choosing a title ensures that it will closely match the content of your essay. Research papers don't always take on the shape we expect them to, and it's easier to match your title to your essay than vice-versa.
- Read through your paper to identify and rework sentences or paragraphs that are confusing or unclear. Each section of your paper should have a clear focus and purpose; if any of yours seem not to meet these expectations, either rewrite or discard them.
- Review your works cited section (at the end of your essay) to ensure that it conforms to the standards of your chosen or assigned style format. You should at least make sure that the style is consistent throughout this section.
- Run a spell checker on your entire document to catch any spelling or grammar mistakes you may not have noticed during your read-through. All modern word processing programs include this function.
- Revise your draft. The first draft of your paper should not be the version you turn in to your teacher (unless you are really good at editing as you write). It is usually necessary to rewrite or at least reorganize and tidy up a research essay before it can be considered finished. Give yourself a day or so away from your first draft to get some distance from your project before going back to revise it.[9] [10]
- Note that revising your draft is not the same as proofreading it. Revisions are done to make sure the content and substantive ideas are solid; editing is done to check for spelling and grammar errors. Revisions are arguably a more important part of writing a good paper.
- You may want to have a friend, classmate, or family member read your first draft and give you feedback. This can be immensely helpful when trying to decide how to improve upon your first version of the essay.
- Except in extreme cases, avoid a complete rewrite of your first draft. This will most likely be counterproductive and will waste a lot of time. Your first draft is probably already pretty good -- it likely just needs some tweaking before it is ready to submit.
- Avoid use of the word "I" in research essay writing, even when conveying your personal opinion about a subject. This makes your writing sound biased and narrow in scope.
- Never plagiarize the work of others! Passing off others' writing as your own can land you in a lot of trouble and is usually grounds for failing an assignment or class.
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| How to Tell if a Calf Muscle is Ruptured Posted: 05 May 2016 01:00 AM PDT Athletes or outdoors enthusiasts may injure their legs at some point during their activities. These injuries are often minor and clear up in a couple of days, but there are injuries that can take longer to heal. A more serious injury is a calf rupture. EditInitially Evaluating a Calf Injury - Understand that a calf rupture is a very common athletic injury.[1] It occurs most frequently in people who are intermittent high-intensity athletes; some doctors refer to these people as the "weekend warriors." This is because suddenly exposing your body to intense exercise, particularly exercise that works your calf muscles, can cause them to tear if you either injure yourself or significantly overwork the muscle.
- One of the most common sports that can result in a calf muscle rupture is tennis, due to the fast stops and starts and the way in which you use your legs.[2] However, a variety of other athletic endeavors can lead to a ruptured calf muscle as well.
- Recognize symptoms of a possible ruptured calf muscle. This can help you to determine whether you may have this injury. Common symptoms include:[3]
- An audible "pop" at the time of the injury.
- Pain in the calf, most often radiating down toward the ankle and foot.
- Pain in the calf that often worsens with motion of the ankle.
- Sudden reduced mobility, up to and including an inability to pivot the foot.
- Look for common physical signs of a ruptured calf muscle. There are some common physical findings that can help you to tell whether or not you may have ruptured your muscle. These include:[4]
- Swelling of the leg around the area of the injured calf muscle; the swelling often extends all the way down to the foot.
- Discolouration due to bruising in the swollen area over the calf.
- Pain upon touching the injured calf area.
- Difficulty or inability to walk on the foot and stand on tiptoes.
EditDiagnosing a Calf Muscle Rupture - See your doctor. The majority of calf muscle ruptures can be identified just telling your doctor what happened and allowing him to quickly examining your leg. Additional tests are not usually necessary, especially since this injury is common and doctors are skilled at recognizing it.[5]
- Have further tests done to exclude other diagnoses. Your doctor may suggest blood tests, X-rays, or other imaging tests to ensure that nothing more serious is going on. Sometimes, other conditions such as deep vein thrombosis (DVT) have the same symptoms as a ruptured calf muscle.[6]
- Ask your doctor how severe your rupture is.[7] If you have been diagnosed with a rupture of your calf muscle, there are three "grades" (or severities) of injury. Your doctor will let you know how bad yours is.
 - A Grade One injury usually involves 10% or less of your muscle. Your muscle will likely be tight and achy for two to five days after the injury. It should start to feel better after that, and you can return to sports when there is no more pain.
- A Grade Two injury usually involves 10 – 90% of the muscle's fibres. You will probably feel considerable pain while walking, and will have felt a sharp pain in the back of your lower leg when the injury occurred. With this degree of injury, you will see swelling and bruising of the calf. Pain will also become worse when the foot is pointed away from your body. Residual tightness and soreness after the swelling goes down can last for a week or more.
- A Grade Three injury is defined as a "full rupture." You will notice severe swelling and bruising in the calf. You may see a bulge toward the top of your calf or right below your knee in cases of a full tear. At the time of the injury, you will have experienced extreme sudden pain in the lower leg. You will likely not be able to contract the muscle in your calf, or use your calf or foot in any way.
- Follow your doctor's suggestions for treatment and optimal healing.[8] In the initial stages of healing (the first two to three days), follow the "RICE" protocol. This stands for Rest, Ice, Compression, and Elevation. Icing your calf muscle is important to diminish swelling, as well as elevating your leg whenever possible (which also helps to diminish swelling). If you place the ice pack inside a tensor bandage or other wrap it can also help with compression.[9] However, be careful not to cut off the circulation in your leg by wrapping it too tightly. When wrapping, be sure that it is gentle and not causing you additional pain.
- As far as rest is concerned, you will want to be gentle on your leg but also continue to use it. In other words, sports or other strenuous activities are not recommended, but gentle walking (if tolerated) can help your calf to heal better. If you cannot yet walk on it, even gentle ankle exercises can promote better healing. Speak to your doctor for specific suggestions.[10]
- Ice for 15 – 20 minutes at a time. Use a cold pack or a bag of ice cubes, and gently wrap it in a cloth if it is too cold on your skin initially.
- If you can elevate your leg on an ottoman or on another chair that is at the same height as the one you are sitting on that helps to reduce swelling. Elevating your leg is not a "make or break" essential of healing; however, if you are sitting down anyway, finding a chair or ottoman to rest your injured leg on can help diminish swelling and is a worthwhile thing to do.
- Take pain medications as needed. Talk to your doctor about what pain medications are safe for you — often something like Advil (Ibuprofen), or Tylenol (Acetaminophen), can help to alleviate pain while your calf muscle is healing.[11] The benefit of something like Advil, as opposed to Tylenol, is that it has anti-inflammatory benefits as well as combatting pain.
- Use a brace if recommended by your doctor. Depending upon the severity of your injury, your doctor may recommend bracing your leg and ankle in a way that allows for the best healing.[12] Do not try doing this on your own; only do it under the guidance of an experienced physician or physiotherapist.
- Always follow up with your doctor and/or physiotherapist regarding a plan for returning to activities and sports.[13] It is important to wait until your injury has fully healed before being too vigorous.
- A gentle squeeze of the calf (soleus) will tell if it is the achilles tendon that has been ruptured along with structures at the bottom of the calf. If the foot will not move with the calf squeeze, the tendon is ruptured and the entire lower leg should be immobilized prior to transport to ensure the best recovery.
- See a doctor immediately if you experience severe lower leg pain. Although you may be sure it is a calf strain, life-threatening conditions such as deep-vein thrombosis (blood clot) can also cause this type of pain.[14]
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