Pauls online notes

Table of Contents Preface Euclidean n-Space

It's amazing to think how much welfare a single person's free work can add to the world. Basically FOSS but for learning math. Apparently many others benefitted as well. I had the honor of being a student of his. I was a high school student while attending college there thanks to a Texas program for gifted kids.

Pauls online notes

Welcome to my online math tutorials and notes. In other words, they do not assume you've got any prior knowledge other than the standard set of prerequisite material needed for that class. The assumptions about your background that I've made are given with each description below. I'd like to thank Shane F, Fred J. I've tried to proof read these pages and catch as many typos as I could, however it just isn't possible to catch all of them when you are also the person who wrote the material. Fred, Mike and David have caught quite a few typos that I'd missed and been nice enough to send them my way. Thanks again Fred, Mike and David! If you are one of my current students and are here looking for homework assignments I've got a set of links that will get you to the right pages listed here. I've made most of the pages on this site available for download as well. These downloadable versions are in pdf format.

Interchange rows i and j Interchange rows j and i Add c times row i to row j Add -c times row i to row j Now that weve got inverse operations we can give the following theorem, pauls online notes. Also, the first two columns represent coefficients of unknowns and so well have two unknowns while the third column consists of pauls online notes constants to the right of the equal sign. Example 2 For the following matrices perform the indicated operation, if possible.

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In this chapter we will start looking at the next major topic in a calculus class, derivatives. This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function. Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative.

Pauls online notes

In the previous section we spent some time getting familiar with series and we briefly defined convergence and divergence. We do, however, always need to remind ourselves that we really do have a limit there! If the sequence of partial sums is a convergent sequence i. Likewise, if the sequence of partial sums is a divergent sequence i.

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Here is this operation. Note as well that in the case of the third scalar multiple we are going to consider the scalar to be a positive. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. As was pointed out in this example there are many paths we could take to do this problem. If the size of a column or row zero matrix is important we will sometimes subscript the size on those as well just to make it clear what the size is. This will sometimes lead to some messy fractions. Okay, at this point, lets stop and insert these into the product so we can make sure that weve got our bearings. Now lets think about the product BC. Notice as well that in several of the steps above we took advantage of the form of several of the rows to simplify the work somewhat and in doing this we did several of the steps in a different order than weve done to this point. We know from Theorem 4 that this is the same as if wed applied the inverse operation to E, but we also know that inverse operations will take an elementary matrix back to the original identity matrix. Solution There is not really a whole lot to do here other than use the definition given above. Once weve looked at solving systems of linear equations well move into the basic arithmetic of matrices and basic matrix properties.

In the introduction to this section we briefly discussed how a system of differential equations can arise from a population problem in which we keep track of the population of both the prey and the predator.

This means that the leading 1 in the second row must be at b2 2 , the leading 1 in the third row must be at b33 , etc. I wanted to take Price too; some people didn't like him, but he's a legend. Example 5 Solve the following system of linear equations. Now, it can be shown that provided we avoid interchanging rows the elementary row operations that we needed to reduce A to U will all have corresponding elementary matrices that are lower triangular matrices. Lets find the second row first. As we go through the steps in this first example well mark the entry s that were going to be looking at in each step in red so that we dont lose track of what were doing. When dealing with real numbers the order in which we write a product doesnt affect the actual result. We will often need to refer to specific entries in a matrix and so well need a notation to take care of that. Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each. Hasz on Nov 13, prev next [—]. Or, they are the negative of the multiple of the first row that we added onto that particular row to get that entry to be zero.