tailieunhanh - Ebook Differential equations and linear algebra (3th edition): Part 2

(BQ) Part 2 book "Differential equations and linear algebra" has contents: Vector spaces and subspaces, eigenvalues and eigenvectors, applied mathematics and ata, fourier and laplace transforms. | Linear Transformations There is nothing mysterious as some have tried to maintain about the applicability of mathematics. What we get by abstraction from something can be returned. R. L. Wilder In the preceding chapter we began building a general framework for studying linear problems. This framework was the mathematical concept of a vector space. Heretofore our attention at any given moment has been focused on the properties of a single vector space V. However a rich mastery of linear algebra also requires a working knowledge of the relationships between different vector spaces. Many problems in linear algebra involve die simultaneous consideration of more than one vector space. For example let A be an tn X n matrix and consider the linear system Ax 0. hl this case note that X is a vector in R while the right-hand-side vector 0 is a vector in Rw. In fact the matrix A can be viewed as a mapping T that accepts inputs X from the vector space V R and yields outputs T x Ax in the vector space w In terms of this mapping the solution set to the homogeneous linear system Ax 0. for example consists of all vectors X in Rn with the property that T x 0. We can use the general mapping notation T in other problems as well. For example consider the second-order differential equation y y o and the associated mapping of functions T defined by T y y y. 342 Definition of a Linear Transformation Given a function y T maps y to the function y y. For example. 7 A 1 2 3 4 A2 -bCv2 2 4-A 2 1 T lnx Inx 4- In A 4- Inx. X2 In terms of the mapping T the solution set s to the differential equation consists of all those functions y that are mapped to the zero function s y T y 0 . The point that we are making is that a variety of problems we have studied to tills point in the text both in linear algebra and in differential equations can be viewed as special cases of the general problem of finding all vectors V in a vector space with the property that T v 0 where T is a mapping .