Mathematics 21bSpring 2005The subject: This is a course on linear algebra and differential equations with a specialsection that also introduces statistical techniques as used in the sciences. As everyone inthe course will be learning linear algebra and learning about differential equations, anintroductory digression is in order to explain what these subjects are about and why theyshould be understood by practicing scientists. These discussions constitute Parts 1 and 2of the digression. The digression has a third part that says some things about the role ofstatistics in the sciences. Part 1: The digression starts with the following definition of science:The purpose of science is to predict future behavior from present circumstance.Here is a somewhat simplistic elaboration: The goal of a scientific program is to predict the outcome of experimentsdone in the future from some prescribed amount of presently known data.For example, Watson and Crick’s original proposal for genetic inheritance assertsthat knowledge of the sequence of bases (adenine, guanine, cytosine, or thymine) alongthe DNA in a living cell is sufficient to predict the sorts of proteins that the cell canproduce. Watson and Crick made this proposal based on their understanding fromexperiments of the workings of a cell, and then further experimental work subsequentlyverified that their proposal is fundamental for understanding how cells encode theirintrinsic operating instructions. By the way, note how theory and experiment work hand in hand: Experiment tellsus the present state of the world, theory provides the prediction for the future, and futureexperiments either falsify the theory or are consistent with its predictions. In this regard,a scientific theory must be falsifiable. To a first approximation, a scientific theory can be viewed in the followingabstract manner: The known data constitutes a labeled collection of numbers, x ={x1, . . ., xn}; here and below, integer subscripts play the role of labels. Meanwhile, datathat arises from future experiments can always be labeled so as to constitutes a secondordered set of numbers, {y1, . . . , yN}. Of course, the latter are not known until theexperiments are carried out. Granted this notation, a theory in science must give a well defined andreproducible method for predicting the future data, {y1, . . ., yN}, from the initial data,{x1, . . ., xn}. Thus, a theory can be viewed as a function that assigns an orderedcollection of N numbers (the y’s) to the collection of n numbers (the x’s). Of course, these experiments, once performed, provide a labeled set of N real datavalues, {y1real, ··· , yNreal}; and if each ykreal is close to its predicted value, yk, then thetheory can be said to be an accurate description of reality. Of course, if some ykreal is farfrom its prediction, yk, the theory needs some revising.The simplest non-constant functions are the linear functions; these have theschematic formy1 = a11 x1 + a12 x2 + · · · a1n xn···yN = aN1 x1 + aN2 x2 + · · · aNn xnwhere the collection {aij}1≤i≤N,1≤≤n are numbers. Thus, a scientific theory that is based onsuch a linear function would have to specify the collection {aij}1≤i≤N,1≤≤n and thenknowledge of the input data {x1, . . ., xn} predicts the output, {y1, . . ., yN}, of the futureexperiments using the preceding equation. As it turns out, the linear functions are among the most relevant to the sciences.This is because an appropriate linear function usually provides the mathematicalequivalent of a ‘first approximation’ for a predictive description of any given phenomena.In any event, a good grasp of the mathematics of linear functions is a prerequisite forfurther explorations because the techniques that are used to study more complicatedfunctions employ most of the mathematics for linear functions.The subject of Linear algebra concerns the mathematics of linear functions. Part 2: It is often the case that the quantity of interest in an experiment can beviewed as a function of some auxiliary variable. Indeed, consider the case when thequantity of interest changes with time, and so can be viewed as a function of time. Thinkof time as a variable, t, that can take values on the real line, and then the quantity ofinterest at time t can be written as a function of t. This is to say that there is a function ofone variable, t u(t), and the values of this function at time t are defined to be those ofthe quantity of interest. Here is a hypothetical example: Radioactive iodine isinadvertently dumped in a reservoir at time t = today. The concentration of iodine in thereservoir at any given future time can be called u(t), and so the assignment t u(t)defines a function of time. In any case, a full theoretical understanding of the time varying behavior of whatis represented abstractly by one or several functions of time entails predicting their valuesat future times from present data. In the reservoir example, the present data might consistof the concentration of iodine measured today and, in addition, the average rates ofinflow, outflow and evaporation of water from the reservoir. As it turns out, theoretical predictions for the future behavior of real worldquantities are often expressed via a system of equations that relate the rate of change ofthe quantities of interest at any given time to the values of the quantities at that sametime. For example, such a theory for predicting the time dependence of some quantitythat is modeled by a function t u(t) would have the form dudt = f(u) ,where the function f is specified by the theory. Such an equation is a simple example ofa differential equation. In general, a differential equation can be said to be any equationfor a function or set of functions that constrains the functions and their derivatives insome specified manner.The subject of differential equations concerns techniques for finding solutions todifferential equations. The subject also concerns techniques for estimating properties ofinterest of hypothetical solutions without knowledge of their explicit form. Linear algebra enters the differential equation story in
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