Download 2.8 Notes 2.9 Exercises

2.8
Notes
The development in Sections 2.4 and 2.5 follows closely that of [9], although the separation
results are all stated positing existence of core points, whereas we prefer to state them in
terms of intrinsic core points, which is slightly more general. The codimension approach
to defining hyperplanes was taken from [4] along with the example of a convex set which
cannot be separated from the origin in Example 2.4.8 (although our proof of this fact is
new). The material in the remaining sections is fairly standard, with [2] being a great
reference in this area. Some of the material in Section 2.5 can also be found in [12].
2.9
Exercises
Exercise 2.1. Prove Proposition 2.1.2: Let B be a Hamel basis of vector space X. Show
that every vector x ∈ X can be written as uniquely as a linear combination of vectors
arising from B.
Exercise 2.2. Let X be a vector space and X 0 its algebraic dual. Show that X 0 is itself
a vector space.
Exercise 2.3. Prove Proposition 2.2.3: If X is a n-dimensional real vector space, then X
is isomorphic (there is a one-on-one and onto vector space homomorphism) to <n .
Exercise 2.4. Prove Proposition 2.1.4: Let M be a subspace of vector space X. Then M
has a complementary subspace.
Exercise 2.5. Let X be an arbitrary vector space. Let X 0 denote the dual of X and X 00
denote the dual of the dual, that is the second dual.
a. Show that X is isomorphic to a subspace of X 00 .
b. Show that if X has finite dimension, then X ∼
= X 00 . In general, if X ∼
= X 00 , then X is
said to be reflexive.
Exercise 2.6. Assume X is a vector space of dimension n and Y is a vector space of
dimension m. Let hom(X, Y ) denote the set of all homomorphisms from X into Y .
a. Prove that hom(X, Y ) is a vector space.
b. What is the dimension of hom(X, Y )? Provide a proof of your answer.
Exercise 2.7. Let S n denote the set of all symmetric matrices. Then S n is a vector space
under matrix addition and scalar multiplication. What is the dimension of this vector
space? Provide a proof of your answer.
Exercise 2.8. Let A = x0 + M be an affine subspace. Let x1 ∈ A then A = x1 + M .
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