Math 4310, Fall 2025
Homework 2
Due by 11pm, Sep 12, 2025
Note: In problems below (and generally in this course), unless stated otherwise, you should always explain
why your answer is correct with an argument or a counterexample as necessary. Simply writing down an
answer, even if correct, will not be worth much.
1. Let ∞ and −∞ be two distinct objects, neither of which is in R. Define an addition and scalar
multiplication on R ∪ {−∞, ∞, } as follows: the sum and product of two real numbers is as usual, and
for α ∈ R, define


if α < 0
 −∞ if α < 0
 ∞
0
if α = 0
0
if α = 0
α∞ =
α(−∞) =


∞
if α > 0
−∞ if α > 0
and
α+∞=∞+α=∞
α + (−∞) = (−∞) + α = −∞
∞+∞=∞
(−∞) + (−∞) = −∞
∞ + (−∞) = (−∞) + ∞ = 0.
With these operations of addition and scalar multiplication, is R ∪ {−∞, ∞} a vector space over R?
Explain.
2. For each of the following vector spaces and each of the specified subsets, determine whether or not the
subset is a subspace. That is, in each case, either verify the conditions defining a subspace (or use the
subspace test), or show by an example that one of the conditions does not hold.
(a) Is {(a, b, c) ∈ R3 : abc = 0} a subspace of R3 ?
(b) Is {(a, b, c) ∈ R3 : a + b = 1} a subspace of R3 ?
(c) Is {(a, b, c) ∈ C3 : a3 = b3 } a subspace of C3 ?
(d) Is {(a, b, c) ∈ F37 : a = [0] or b = [0]} a subspace of F37 ?
3. Let V = R3 and let U = {(a, b, c) ∈ V |a + b + c = 0}. For each of the following subspaces W , state
whether or not V = U ⊕ W . You do not have to prove this.
(a) W = {(a, 0, −a) : a ∈ R}
(b) W = {(a, 0, a) : a ∈ R}
(c) W = {(a, b, c) ∈ V : a = c}
4. Let C(R) be the real vector space consisting of all continuous functions R → R. A function f : R → R
is called even if
f (−x) = f (x), for all x ∈ R.
A function f : R → R is called odd if
f (−x) = −f (x), for all x ∈ R.
Let Ve be the set of functions f ∈ C(R) that are even, and let Vo be the set of functions f ∈ C(R)
that are odd. The sets Ve and Vo are subspaces of C(R) (you do not have to prove this). Prove that
C(R) = Ve ⊕ Vo .
2