Linear Algebra, 2020a

Section III. Basis and Dimension 135
2.36 A basis for a space consists of elements of that space. So we are naturally led to
how the property ‘is a basis’ interacts with operations ⊆ and ∩ and ∪. (Of course,
a basis is actually a sequence that it is ordered, but there is a natural extension of
these operations.)
(a) Consider first how bases might be related by ⊆. Assume that U, W are
subspaces of some vector space and that U ⊆ W. Can there exist bases B U for U
and 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 .)
̌ 2.37 Consider how ‘dimension’ interacts with ‘subset’. Assume U and W are both
subspaces 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.
2.38 Here is an alternative proof of this section’s main result, Theorem 2.4. First is
an example, then a lemma, then the theorem.
(a) Express this vector from R 3 a as a linear combination of members of the basis.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 0
1
B = 〈 ⎝ ⎠ , ⎝ ⎠ , ⎝ ⎠〉 ⃗v = ⎝ ⎠
0
0
1
0
(b) In that combination pick a basis vector with a non-zero coefficient. Alter B
by exchanging ⃗v for that basis vector, to get a new sequence ˆB. Check that ˆB is
also a basis for R 3 .
(c) (Exchange Lemma) Assume that B = 〈⃗β 1 ,...,⃗β n 〉 is a basis for a vector space,
and that for the vector ⃗v the relationship ⃗v = c 1
⃗β 1 + c 2
⃗β 2 + ···+ c n
⃗β n has c i ≠ 0.
Prove that exchanging ⃗v for ⃗β i yields another basis for the space.
(d) Use that, with induction, to prove Theorem 2.4.
? 2.39 [Sheffer] A library has n books and n + 1 subscribers. Each subscriber read at
least one book from the library. Prove that there must exist two disjoint sets of
subscribers who read exactly the same books (that is, the union of the books read
by the subscribers in each set is the same).
? 2.40 [Wohascum no. 47] For any vector ⃗v in R n and any permutation σ of the
numbers 1, 2, ..., n (that is, σ is a rearrangement of those numbers into a new
order), define σ(⃗v) to be the vector whose components are v σ(1) , v σ(2) , ..., and
v σ(n) (where σ(1) is the first number in the rearrangement, etc.). Now fix ⃗v and let
V be the span of {σ(⃗v) | σ permutes 1, ..., n}. What are the possibilities for the
dimension of V?
0
2
2
0