Linear Algebra, 2020a

106 Chapter Two. Vector Spaces
(c) Show that any subspace of R 3 must pass through the origin, and so any
subspace of R 3 must involve zero in its description. Does the converse hold?
Does any subset of R 3 that contains the origin become a subspace when given
the inherited operations?
2.39 We can give a justification for the convention that the sum of zero-many vectors
equals the zero vector. Consider this sum of three vectors ⃗v 1 +⃗v 2 +⃗v 3 .
(a) What is the difference between this sum of three vectors and the sum of the
first two of these three?
(b) What is the difference between the prior sum and the sum of just the first
one vector?
(c) What should be the difference between the prior sum of one vector and the
sum of no vectors?
(d) So what should be the definition of the sum of no vectors?
2.40 Is a space determined by its subspaces? That is, if two vector spaces have the
same subspaces, must the two be equal?
2.41 (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.
2.42 Show that the span of a set of vectors does not depend on the order in which
the vectors are listed in that set.
2.43 Which trivial subspace is the span of the empty set? Is it
⎛ ⎞
0
{ ⎝0⎠} ⊆ R 3 , or {0 + 0x} ⊆ P 1 ,
0
or some other subspace?
2.44 Show that if a vector is in the span of a set then adding that vector to the set
won’t make the span any bigger. Is that also ‘only if’?
̌ 2.45 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 a
subspace? Always? Sometimes? Never?
(b) Must the union A ∪ B be a subspace?
(c) If A is a subspace of some V, must its set complement V − A be a subspace?
(Hint. Try some test subspaces from Example 2.19.)
2.46 Does the span of a set depend on the enclosing space? That is, if W is a
subspace of V and S is a subset of W (and so also a subset of V), might the span
of S in W differ from the span of S in V?
2.47 Is the relation ‘is a subspace of’ transitive? That is, if V is a subspace of W
and W is a subspace of X, must V be a subspace of X?
2.48 Because ‘span of’ is an operation on sets we naturally consider how it interacts
with 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]?