Linear Algebra

96 Chapter Two. Vector Spaces(b) What is the difference between the prior sum and the sum of just the firstone vector?(c) What should be the difference between the prior sum of one vector and thesum of no vectors?(d) So what should be the definition of the sum of no vectors?2.38 Is a space determined by its subspaces? That is, if two vector spaces have thesame subspaces, must the two be equal?2.39 (a) Give a set that is closed under scalar multiplication but not addition.(b) Give a set closed under addition but not scalar multiplication.(c) Give a set closed under neither.2.40 Show that the span of a set of vectors does not depend on the order in whichthe vectors are listed in that set.2.41 Which trivial subspace is the span of the empty set? Is it⎛ ⎞0{ ⎝0⎠} ⊆ R 3 , or {0 + 0x} ⊆ P 1 ,0or some other subspace?2.42 Show that if a vector is in the span of a set then adding that vector to the setwon’t make the span any bigger. Is that also ‘only if’?̌ 2.43 Subspaces are subsets and so we naturally consider how ‘is a subspace of’interacts with the usual set operations.(a) If A, B are subspaces of a vector space, must their intersection A ∩ B be asubspace? Always? Sometimes? Never?(b) Must the union A ∪ B be a subspace?(c) If A is a subspace, must its complement be a subspace?(Hint. Try some test subspaces from Example 2.19.)̌ 2.44 Does the span of a set depend on the enclosing space? That is, if W is asubspace of V and S is a subset of W (and so also a subset of V), might the spanof S in W differ from the span of S in V?2.45 Is the relation ‘is a subspace of’ transitive? That is, if V is a subspace of Wand W is a subspace of X, must V be a subspace of X?̌ 2.46 Because ‘span of’ is an operation on sets we naturally consider how it interactswith the usual set operations.(a) If S ⊆ T are subsets of a vector space, is [S] ⊆ [T]? Always? Sometimes?Never?(b) If S, T are subsets of a vector space, is [S ∪ T] = [S] ∪ [T]?(c) If S, T are subsets of a vector space, is [S ∩ T] = [S] ∩ [T]?(d) Is the span of the complement equal to the complement of the span?2.47 Reprove Lemma 2.15 without doing the empty set separately.2.48 Find a structure that is closed under linear combinations, and yet is not avector space. (Remark. This is a bit of a trick question.)