Page 25 - Linear Algebra Done Right

Properties of Vector Spaces
Properties of Vector Spaces
The definition of a vector space requires that it have an additive 11
identity. The proposition below states that this identity is unique.
1.2 Proposition: A vector space has a unique additive identity.

Proof: Suppose 0 and 0 are both additive identities for some vec-

tor space V. Then
0 = 0 + 0 = 0,

where the first equality holds because 0 is an additive identity and the
second equality holds because 0 is an additive identity. Thus 0 = 0,

proving that V has only one additive identity. The symbol means
“end of the proof”.
Each element v in a vector space has an additive inverse, an element
w in the vector space such that v +w = 0. The next proposition shows
that each element in a vector space has only one additive inverse.

1.3 Proposition: Every element in a vector space has a unique
additive inverse.

Proof: Suppose V is a vector space. Let v ∈ V. Suppose that w
and w are additive inverses of v. Then

w = w + 0 = w + (v + w ) = (w + v) + w = 0 + w = w .

Thus w = w , as desired.

Because additive inverses are unique, we can let −v denote the ad-
ditive inverse of a vector v. We define w − v to mean w + (−v).
Almost all the results in this book will involve some vector space.
To avoid being distracted by having to restate frequently something
such as “Assume that V is a vector space”, we now make the necessary
declaration once and for all:

Let’s agree that for the rest of the book
V will denote a vector space over F.