Review Sheet for Final Exam - Differential Equations and Linear Algebra | MATH 2250, Exams of Mathematics

Download Review Sheet for Final Exam - Differential Equations and Linear Algebra | MATH 2250 and more Exams Mathematics in PDF only on Docsity! Review Sheet Math 2250-3 November 10, 2004 Our exam covers chapters 4-5 of the text. Only scientific calculators will be allowed on the exam. But you can expect to be working with Maple output, in ways consistent with the practice exam below and the homework problems you have worked. Chapter 4: At most 40% of the exam will deal directly with this material....but much of Chapter 5 uses these concepts, so beware. Know Definitions: (a) Vector Space: A collection of objects which can be added and scalar multiplied, 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^n a vector space, but also certain subsets of it are, and so are spaces made out of functions...because functions can be added and scalar multiplied (page 265.) (b) 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 (which I called the nullspace of the matrix and which the book calls the solution space, page 243), the span of a collection of vectors (page 248), AND the set of homogenous solutions to a linear differential equation (section 5.2). (c) A linear combination of a set of vectors {v1, v2, ...vn} is any expression c1*v1 + c2*v2 + ... + cn*vn. (page 246) (d) The span of a set of vectors {v1, v2, ...vn} is the collection of all linear combinations. (page 248) (e) A collection {v1, v2, ...vn} is linearly dependent if and only if some linear combinatation (with not all ci’s = 0) adds up to the zero vector. (f) A collection {v1, ... vn} is linearly independent if and only if the only linear combination of them which adds up to zero is the one in which all coefficients ci=0. (page 249) (g) A basis for a vector space (or subspace) is a set of vectors {v1, ..., vk} which space the space and which are linearly independent. (page 255.) (h) The dimension of a vector space is the number of elements in any basis. Know Facts: (a) If the dimension of a vector space is n, then no collection of fewer than n vectors can span and every collection with more than n elements is dependent. (b) n vectors in R^n are a basis if and only if the square matrix in which they are the columns is non-singular. So you can use det or rref as a test for basis in this case. (c) Basically all linear independence and span questions in R^n can be answered using rref. (see below.) (d) You can toss dependent vectors out of a collection without changing the span. In this manner you can take a spanning set and turn it into a basis. Do computations: (a) Be able to check whether vectors are independent or independent, e.g. problems page 248. (4.3) Know how to use rref to check for dependencies. (b) Be able to find bases for the solution space to homogeneous equations, e.g. problems page 255 (4.4) (c) Be able to find bases for rowspace and column space, e.g. problems page 263 (4.5)