Linear Algebra

120 Chapter Two. Vector Spaceš 2.25 Where S is a set, the functions f: S → R form a vector space under the naturaloperations: the sum f + g is the function given by f + g (s) = f(s) + g(s) and thescalar product is r · f (s) = r · f(s). What is the dimension of the space resulting foreach domain?(a) S = {1} (b) S = {1, 2} (c) S = {1, . . . , n}2.26 (See Exercise 25.) Prove that this is an infinite-dimensional space: the set ofall functions f: R → R under the natural operations.2.27 (See Exercise 25.) What is the dimension of the vector space of functionsf: S → R, under the natural operations, where the domain S is the empty set?2.28 Show that any set of four vectors in R 2 is linearly dependent.2.29 Show that 〈⃗α 1 , ⃗α 2 , ⃗α 3 〉 ⊂ R 3 is a basis if and only if there is no plane throughthe origin containing all three vectors.2.30 (a) Prove that any subspace of a finite dimensional space has a basis.(b) Prove that any subspace of a finite dimensional space is finite dimensional.2.31 Where is the finiteness of B used in Theorem 2.5?̌ 2.32 Prove that if U and W are both three-dimensional subspaces of R 5 then U ∩ Wis non-trivial. Generalize.2.33 A basis for a space consists of elements of that space. So we are naturally led tohow the property ‘is a basis’ interacts with operations ⊆ and ∩ and ∪. (Of course,a basis is actually a sequence in that it is ordered, but there is a natural extensionof these operations.)(a) Consider first how bases might be related by ⊆. Assume that U, W aresubspaces of some vector space and that U ⊆ W. Can there exist bases B U for Uand B W for W such that B U ⊆ B W ? Must such bases exist?For any basis B U for U, must there be a basis B W for W such that B U ⊆ B W ?For any basis B W for W, must there be a basis B U for U such that B U ⊆ B W ?For any bases B U , B W for U and W, must B U be a subset of B W ?(b) Is the ∩ of bases a basis? For what space?(c) Is the ∪ of bases a basis? For what space?(d) What about the complement operation?(Hint. Test any conjectures against some subspaces of R 3 .)̌ 2.34 Consider how ‘dimension’ interacts with ‘subset’. Assume U and W are bothsubspaces of some vector space, and that U ⊆ W.(a) Prove that dim(U) dim(W).(b) Prove that equality of dimension holds if and only if U = W.(c) Show that the prior item does not hold if they are infinite-dimensional.? 2.35 [Wohascum no. 47] For any vector ⃗v in R n and any permutation σ of thenumbers 1, 2, . . . , n (that is, σ is a rearrangement of those numbers into a neworder), define σ(⃗v) to be the vector whose components are v σ(1) , v σ(2) , . . . , andv σ(n) (where σ(1) is the first number in the rearrangement, etc.). Now fix ⃗v andlet V be the span of {σ(⃗v) ∣ ∣ σ permutes 1, . . . , n}. What are the possibilities forthe dimension of V?