Please solve problem 5 2 Dual SpacesLet V and W be vector spaces over R. Recall that L(V, W), the set of all linear maps from V to Wis a vector space over R. When W = R, L(V, R) is precisely the set of all linear functionals on V, andhence forms a vector space over R. This space is called the dual space of V. We record this importantfact in the next theorem.Theorem 2.1. Let V be a vector space over R. Then the dual space of V, denoted by V*, is the set of all linearfunctionals on V. The dual space V* is a vector space over R.The next project problem concerns description of the dual space of a finite dimensional vectorspace.Theorem 2.2. Let V be a finite dimensional vector space over R, with basis {01, 02,,n}. Let 9₁, 92, ···, Pn €V* be defined as follows:Pk(k) = 1, and Pk (vj) = 0, if j + k, where k, je {1,2,...,n}.Then {91, 92, 9n} forms a basis for V*. Hence, dim(V) = dim(V*).Project Problem 4: Prove the above Theorem 2.2Project Problem 5: Use Theorem 2.2 to prove the following statement. Consider V = R", and leto be a linear functional on R". Then p() = a1x₁ +anxn, for some fixed a1, a2,,an R, where7 = (x1,...,xn).Project Problem 6: Use Theorem 2.2 to prove the following statement. Consider V = M,, (R), andlet q be a linear functional on M, (R). Then we can find an n x n matrix H, such that p(X) = Trn (HX)for all X € M₂ (R). Here HX denotes the matrix multiplication of the nx n matrices H and X.Hint: The (i, j)th entry of H is given by p(Eji), where Eij is the n x n matrix, whose (i, j)th entry is1, and rest of the entries are 0. Recall that E1,1,, E1,2,En,n forms a basis for M,, (R).Final Remarks: The dual space of a vector space is a very important notion to understand thevector space Vitself. The study of dual spaces of infinite dimensional vector spaces is quite challengingin general, and is one of the main focus of Functional Analysis.

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2 Dual Dual Spaces Let V and W be vector spaces over R. Recall that L(V, W), the set of all linear maps from V to W is a vector space over R. When W = R, L(V, R) is precisely the set of all linear functionals on V, and hence forms a vector space over R. This space is called the dual space of V. We record this important fact in the next theorem. Theorem 2.1. Let V be a vector space over R. Then the dual space of V, denoted by V*, is the set of all linear functionals on V. The dual space V* is a vector space over R. The next project problem concerns description of the dual space of a finite dimensional vector space. Theorem 2.2. Let V be a finite dimensional vector space over R, with basis (0₁, 02,..., n. Let 9₁, 92,..., On €

2 Dual Dual Spaces Let V and W be vector spaces over R. Recall that L(V, W), the set of all linear maps from V to W is a vector space over R. When W = R, L(V, R) is precisely the set of all linear functionals on V, and hence forms a vector space over R. This space is called the dual space of V. We record this important fact in the next theorem. Theorem 2.1. Let V be a vector space over R. Then the dual space of V, denoted by V, is the set of all linear functionals on V. The dual space V is a vector space over R. The next project problem concerns description of the dual space of a finite dimensional vector space. Theorem 2.2. Let V be a finite dimensional vector space over R, with basis (0₁, 02,..., n. Let 9₁, 92,..., On €

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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