Diff Equ & Lin Algebra
Diff Equ & Lin Algebra MATH 2250
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This 2 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 2250 at University of Utah taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/229947/math-2250-university-of-utah in Mathematics (M) at University of Utah.
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Date Created: 10/26/15
Review of Chapter 4 and 5 for Exam 2 Math 2250 2 Summer 2007 Exam 2 covers materials from sections 41 44 and 51 55 of the text Chapter 4 Important de nitions to know a Vector Space A collection of objects which can be added and scalar multiplied7 so that the usual arithemetic properties page 240 hold You do not need to memorize all eight of these properties The key point is that not only is R a vector space7 but also certain subsets of it are7 and so are spaces made out of functions because functions can be added and scalar multiplied page 265 A T V Subspace a subset of a vector space which is itself of vector space To check whether a subset is actually a subspace you only have to show that sums and scalar multiples of subset elements are also in the subset Theorem 1 page 242 Examples of important subspaces are the set of homogeneous solutions to a matrix equation Ax0 page 2437 the span of a collection of vectors page 2487 and the set of homogenous solutions to a linear differential equation section 52 A O V A linear combination of a set of vectors v17vz7 vn is any expression clvl 3sz cnvn for some scalar coefficients 0102 7 on page 246 d The span of a set of vectors v17 vz7 vn is the collection of all linear combinations page 248 e A collection v17 v27 vn is linearly independent if and only if the only solution to clvl 3sz cnvnOisc1cgcn0 If there is a set 0102 70 not all zeros such that solves the equation then the vectors are linearly dependent page 249 f A basis for a vector space or subspace is a set of vectors v17 vz7 vn which span the space and which are linearly independent page 255 g The dimension of a vector space is the number of elements in any basis Useful Facts a If the dimension of a vector space is n7 then no collection of fewer than n vectors can span and every collection with more than n elements is dependent For example7 you need at least n vectors to span R A set of n linearly independent vectors spans R b n vectors in R forms a basis if and only if the square matrix that results from putting the vectors as columns is non singular So you can use the determinant or RREF as a way to test for span andor linear independence Make sure you know how to do these computations a Be able to check whether vectors are independent or independent7 eg problems on page 248 in 43 Know how to use RREF to check for linear independence and write down dependencies if the vectors are not linearly independent b Be able to nd bases for the solution space to homogeneous equations Ax 07 eg problems on page 255 in 44 Chapter 5 a Two functions are linearly independent if they are not constant multiples of each other One way A O V A D to check for linear independence is to use the Wronskian Make sure you know how to check linear independence given 71 functions7 f1z7 f2z7 f3z7 fnm7 by showing that their Wronskian is not equal to O Homogeneous linear n th order equation pnzy pn71zy 1 p1zy Poy 0 General solution consists of n linearly independent solution That is7 all possible solutions to this equa tion can be writeen as a linear combination of 71 functions ie the solution space is an n dimensional vector space Be able to solve for a speci c solution when initial conditions are given Homogeneous equation with constant coef cients any M71240 112 10990 0 Know how to nd a general solution consisting of exponentials using characteristic equations What to do when the root is complex What about when you have repeated roots Non hom ogeneous linear equation 1 The general solution consists of a complemetary solution which solves the corresponding homogeneous problem and a particular solution Be able to solve a non homogeneous equation for equations with constant coef cients using meth ods of undetermined coef cients Know what to do when x is not linearly independent to the complementary solution pnxy pn71xy 1 p1y 1909090 Application mechanical vibrations Only unforced system will be tested in Exam 2 mm 0 km 0 Make sure you are able to derive this equation from Newtons and Hookes Laws Know what the solution is and what it means for the different cases 0 undamped motion 0 0 Be able to go from Acoswt Bsinwt to Ccoswt 7 04 using the ABC triangle to nd the amplitude C and phase 04 o damped case 0 7 0 There are three possible cases under damped7 over damped7 critically damped Know how to recognize7 and different forms of the solution
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