Mathematics 369
Homework (due Feb 1)
A. Hulpke
7) Let V be a vector space over F and 0 ∈ V the zero vector. Show that for every c ∈ F we
have that c · 0 = 0.
8) Let x, y, z be elements of a vector space V . Show that if x + y = x + z then y = z.
9∗ ) Let V = {(a : b) | a, b ∈ R} the set of all colon-separated pairs of real numbers. We define an “addition” (this is not the usual componentwise addition, we just call it “addition” for
this problem) on V by setting (a : b) + (c : d) = (a + c) : (b − d) and a scalar “multiplication”
by R on V by α · (a : b) = (αa : αb).
Is V with these operations a vector space? Justify your answer!
10) Which of the following sets are vector spaces over R? Give a proof or a counterexample (Avoid unnecessary work. Be careful which axioms you actually need to check!)?
a) The set of even functions: { f : R → R | f (−x) = f (x)} (with the usual scalar multiplication
and addition of functions).
b) The set C of complex numbers with the usual addition and the usual scalar multiplication
by R.
c) The set of ordered 2-tuples {(a, b)T ∈ R2 | a ≥ b} with componenwise addition and scalar
multiplication by R.
d) The set of solutions to the differential equation f 00 (x) + f 0 (x) − 3 f (x) = 0.
e) The set {(0, 0)T } with componentwise addition and scalar multiplication.
f) The set of nonsingular matrices {A ∈ R2×2 | det(A) 6= 0}.
11) In each of the following cases you are given a vector space V (all of them have been
proven to be a vector space in class) and a subset S ⊂ V . In each case, check whether S is a
subspace of V (in which case we can write S ≤ V ). (Be aware that you only need to check
the closure axioms!) Justify your answer.
a) V = R3 , S = {(1, a, b)T ∈ V | a, b ∈ R}.
b) V = R3 , S = {(a, b, c)T ∈ V | 3a + 4b = 2c}.
c) V = R3 , S = {(a, b, c)T ∈ V | a2 = 1}.
d) V = C[0, 10], S = { f ∈ V | f (5) = 0}.
e) V = C[0, 10], S = { f ∈ V | f (5) = 1}.
f) V = C2 [−∞, ∞], S = { f ∈ V | f 00 + 3 f 0 = 0}.
g) V = P3 , S = { f ∈ V | f (x) = a + bx2 } (i.e. f is a polynomial without an x-term).
Problems marked with a ∗ are bonus problems for extra credit.
Set Notation
In mathematics one often wants to define sets of elements, by selecting some elements from
a larger set A, that fulfill a certain property. (For example: All positive real numbers that are
smaller than 5.) This is done easiest by the following notation:
{a ∈ A | property(a)}
The left part (before the |) indicates that we take elements from a set A. The name a is only
used, as we want to refer to these elements in the right part.
The right part indicates a property, that these elements will have to fulfil, i.e. the new set
only consists of those elements, for which the property is fulfilled.
For example, I could describe the set of students with a blue sweater as:
S1 = {s ∈ Students | s wears a blue sweater}
If R+ is the set of positive real numbers we can denote the positive real numbers smaller
than 5 by
S2 = {a ∈ R+ | a < 5}
or I could just refer to the real numbers R and impose two conditions:
S2 = {a ∈ R | a > 0, a < 5}
Similarly, the set
S3 = { f ∈ C[0, 2] | f (1) = 0}
consists of all functions continuous on the interval [0, 2] that have a zero at 1. This notation
is not unique. We could describe S2 as: {a ∈ R | a > 0, a < 5} or {a ∈ R | 0 < a < 5}.
When working with such sets, one typically will use the specified property. For example if
we use again S3 = { f ∈ C[0, 2] | f (1) = 0}, we can deduce that f (1) = 0 (which definitevely
does not hold for any arbitrary function). On the other hand, if g is an arbitrary function,
we can check that g ∈ S3 by verifying the property, i.e. that g(1) = 0. Thus for example
x − x2 ∈ S3 and ex 6∈ S3 .
(Some people prefer to use a semicolon (;) instead of the line |.)