Given vector spaces VV and WW over FF. Let L(V,W)L(V,W) be the set of all linear maps from VV to WW. With addition and scalar multiplication as given in Definition 3.6 of the textbook, show that L(V,W)L(V,W) satisfies the distributive properties of a vector space. Suppose V and W are finite-dimensional. Then L.V; W / is finitedimensional anddimL(V,W) = D (dim V)( dim W) 84CHAPTER 3 Linear MapsLinear Maps Thought of as Matrix MultiplicationPreviously we defined the matrix of a linear map. Now we define the matrixof a vector.3.62 Definition matrix of a vector, M(v)Suppose v e V and v1,... Vn is a basis of V. The matrix of v withrespect to this basis is the n-by-1 matrixM(v) =Cnwhere c1,..,Cn are the scalars such thatv = c1V1 +.+ CnVn-The matrix M(v) of a vector v e V depends on the basis v1,.., Vn of V,as well as on v. However, the basis should be clear from the context and thusit is not included in the notation.3.63 Example matrix of a vector• The matrix of 2– 7x +5x³ with respect to the standard basis of P3(R)is()2-75• The matrix of a vector x e F" with respect to the standard basis isobtained by writing the coordinates of x as the entries in an n-by-1matrix. In other words, if x = (x1,..., Xn) € F", then()X1M(x) =XnOccasionally we want to think of elements of V as relabeled to be n-by-1matrices. Once a basis v1,..., Vn is chosen, the function M that takes v e Vto M(v) is an isomorphism of V onto F",1 that implements this relabeling. 86CHAPTER 3 Linear MapsBecause the result above allows us to think (via isomorphisms) of eachlinear map as multiplication on F".1 by some matrix A, keep in mind that thespecific matrix A depends not only on the linear map but also on the choiceof bases. One of the themes of many of the most important results in laterchapters will be the choice of a basis that makes the matrix A as simple aspossible.In this book, we concentrate on linear maps rather than on matrices. How-ever, sometimes thinking of linear maps as matrices (or thinking of matricesas linear maps) gives important insights that we will find useful.OperatorsLinear maps from a vector space to itself are so important that they get aspecial name and special notation.3.67 Definition operator, L(V)A linear map from a vector space to itself is called an operator.The notation L(V) denotes the set of all operators on V. In otherwords, L(V) = L(V, V).The deepest and most importantparts of linear algebra, as well asmost of the rest of this book, dealwith operators.A linear map is invertible if it isinjective and surjective. For an op-erator, you might wonder whether in-jectivity alone, or surjectivity alone,is enough to imply invertibility. Oninfinite-dimensional vector spaces, neither condition alone implies invert-ibility, as illustrated by the next example, which uses two familiar operatorsfrom Example 3.4.3.68 Example neither injectivity nor surjectivity implies invertibility• The multiplication by x² operator on P(R) is injective but not surjective.The backward shift operator on F is surjective but not injective.In view of the example above, the next result is remarkable-it statesthat for operators on a finite-dimensional vector space, either injectivity orsurjectivity alone implies the other condition. Often it is easier to check thatan operator on a finite-dimensional vector space is injective, and then we getsurjectivity for free.

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Answered: Suppose V and W are finite-dimensional.…

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Suppose V and W are finite-dimensional. Then L.V; W / is finitedimensional and dimL(V,W) = D (dim V)( dim W)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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