Math 115A Worksheet
Thursday, Oct 12 (Week 2)
Recall: For a eld F , a vector space over F is a set V together with two operations, addition
(V × V → V ) and scalar multiplication (F × V → V ), which satisfy all of the following
properties (axioms):
(VS 0) For all ⃗x, ⃗y ∈ V , ⃗x + ⃗y ∈ V . (V is closed under addition.)
and
For all a ∈ F and all ⃗x ∈ V , a⃗x ∈ V . (V is closed under scalar multiplication.)
(VS 1) For all ⃗x, ⃗y ∈ V ,
⃗x + ⃗y = ⃗y + ⃗x▷
(Addition in V is commutative.)
(VS 2) For all ⃗x, ⃗y, ⃗z ∈ V ,
(⃗x + ⃗y) + ⃗z = ⃗x + (⃗y + ⃗z)▷
(Addition in V is associative.)
(VS 3) There exists an element
⃗
0 ∈ V such that, for all ⃗x ∈ V ,
⃗
0 + ⃗x = ⃗x▷
(There is an additive identity element. Furthermore, we proved it is unique.)
(VS 4) For all ⃗x ∈ V , there exists ⃗y ∈ V such that
⃗x + ⃗y =
⃗
0▷
(Every element has an additive inverse.)
(VS 5) For all ⃗x ∈ V ,
1⃗x = ⃗x▷
(VS 6) For all a, b ∈ F and all ⃗x ∈ V ,
a(b⃗x) = (ab)⃗x▷
(Scalar multiplication is associative.)
(VS 7) For all a ∈ F and all ⃗x, ⃗y ∈ V ,
a(⃗x + ⃗y) = a⃗x + a⃗y▷
(Scalar multiplication is distributive on the left.)
(VS 8) For all a, b ∈ F and all ⃗x ∈ V ,
(a + b)⃗x = a⃗x + b⃗x▷
(Scalar multiplication is distributive on the right.)
You will need the above denition for all of the questions on this worksheet.
1. Last week, we proved the uniqueness of the additive inverse and the multiplicative
inverse for any element of a eld. Here you will do the same for additive inverses in a
vector space.
First, complete the statement of a theorem below about the uniqueness of additive
inverses for elements of a vector space V . (Be careful about the order of quantiers!)
Then prove your theorem as a corollary of the cancellation law for vector addition
(Theorem 1.1 in §1.2).
Theorem. Let F be a eld, and let V be a vector space over F .
Proof.
2. Last week, we proved cancellation laws for addition and multiplication in a eld. There
are two cancellat ion laws for scalar multiplication in a vector space. In this problem,
you will prove the rst of them. Fill in the two blanks in the following statement of
a theorem, and then prove the theorem. In your proof, try to be explicit about every
eld axiom (F [0-5]) and every vector space axiom (VS [0-8]) that you use.
Theorem. Let V be a vector space over a eld F . For all a ∈ F and all ⃗x, ⃗y ∈ V , if
a⃗x = a⃗y and , then .
Proof.
For the next two problems, you will need the following theorem. (This is Theorem 1.2 in
your textbook.)
Theorem. Let V be a vector space over a eld F .
(a) For each ⃗x ∈ V , 0⃗x =
⃗
0.
(b) For each a ∈ F and each ⃗x ∈ V , (−a)⃗x = −(a⃗x) = a(−⃗x).
(c) For each a ∈ F , a
⃗
0 =
⃗
0.
3. Fill in the blank in the following theorem, to get the “zero product property” for vector
spaces. (Recall that we proved the analogous fact for elds in class on Jan 10.) Then
prove the theorem.
Theorem. Let V be a vector space over a eld F . For any a ∈ F and any ⃗x ∈ V , if
a⃗x =
⃗
0, then
.
Proof.
(Hint: Remember how to prove an either-or statement: “P or Q” is logically equivalent
to “If P is false, then Q is true”, and also equivalent to “If Q is false, then P is true”.)
4. In this problem, you will prove the second of the cancellation laws for scalar multipli-
cation in a vector space. Fill in the two blanks in the following statement of a theorem,
and then prove the theorem.
Theorem. Let V be a vector space over a eld F . For all a, b ∈ F and all ⃗x ∈ V , if
a⃗x = b⃗x and , then .
Proof.
(Hint: This one is harder than the other one. But the previous problem will be helpful!)
In the next homework assignment (Homework 3), you will be using the following two impor-
tant denitions in several problems. Here is the rst of those denitions:
Defnition. Let V be a vector space over a eld F , and let X and Y be nonempty subsets
of V . Then the sum of X and Y , denoted X + Y , is the set
{ x + y | x ∈ X and y ∈ Y } ▷
5. In the vector space R
2
, consider the subsets X = {(1, 1), (2, 2), (3, 3)} and Y =
{(1, 0), (0, 1)}. Compute the set X + Y .
(Note that neither of these sets are subspaces of R
2
. This problem is merely an example
to get you used to the idea of what the subset X + Y means.)
6. Recall that two sets are equal if (and only if) they contain exactly the same elements.
Therefore, given two sets A and B, to show that A = B, the standard way is to show
that (1) for all x ∈ A, it is also true that x ∈ B, and (2) for all x ∈ B, x ∈ A as
well. Note that (1) is the same as showing A ⊆ B and (2) is the same as showing that
B ⊆ A.
Let V = R
3
, and let
X = span
(
{(1, 1, 0)}
)
= { (a, a, 0) | a ∈ R } and
Y = span
(
{(1, −1, 0)}
)
= { (a, −a, 0) | a ∈ R } ▷
Let
W
= { (a
1
, a
2
, a
3
) ∈ R
3
| a
3
= 0 }. Prove that X + Y = W .
Here is the second of the two denitions from Homework 3:
Defnition. Let V be a vector space, and let
W
1
and
W
2
be subspaces of V . We say that
V is the (internal) direct sum of W
1
and
W
2
if both of the following are true:
(i) V = W
1
+ W
2
(see the previous denition) and
(ii)
W
1
∩
W
2
= {
⃗
0}.
If V is the direct sum of W
1
and
W
2
, we write V =
W
1
⊕ W
2
.
7. Let V be a vector space over a eld F , and suppose
W
1
and
W
2
are two subspaces of
V such that V =
W
1
⊕
W
2
. Let ⃗x ∈ V . Then by part (i) of the above denition, we
have ⃗x ∈
W
1
+ W
2
, which means that there must exist some ⃗w
1
∈
W
1
and ⃗w
2
∈ W
2
such that ⃗x = ⃗w
1
+ ⃗w
2
. Prove that ⃗w
1
and ⃗w
2
are unique.
(Hint: Remember how to prove something is unique! It will help to rst carefully state
more clearly what is meant by “ ⃗w
1
and ⃗w
2
are unique”.)
8. Once again let V be a vector space over a eld F , and suppose
W
1
and
W
2
are subspaces
of V . This time, suppose that
W
1
+ W
2
= V , but
W
1
∩
W
2
̸= {
⃗
0}. Let ⃗x ∈ V . Once
again, since
W
1
+
W
2
= V , we have ⃗x ∈ W
1
+
W
2
, so there must exist ⃗w
1
∈
W
1
and
⃗w
2
∈
W
2
such that ⃗x = ⃗w
1
+ ⃗w
2
. This time, however, show that ⃗w
1
and ⃗w
2
are not the
unique vectors satisfying this.
