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Chapter 4
Vector Spaces
The vectors in R2 (the plane) and R3 (space) we studied in the previous chapter
are example of a large family of similar objects which arise naturally in many
applications. In this chapter, we generalize the notion of vectors and study their
properties. As we do so, we will start gaining an appreciation for the power of
linear algebra.
4.1
Vector Spaces
We extract the most important properties of vectors studied in the previous
chapter. We turn them into axioms and de…ne a vector space as a set which
satisfy these axioms. More speci…cally, we have the following de…nition.
De…nition 247 (Vector Space) A vector space is a non-empty set of objects V , called vectors, on which are de…ned two operations, called addition
and scalar multiplication which must satisfy the ten axioms listed below. These
axioms must be satis…ed for every vector u, v, w in V and every scalar c and
d.
1. u + v 2 V (V is closed under addition)
2. u + v = v + u (commutative property)
3. (u + v) + w = u + (v + w) (associative property)
4. There exists a vector in V , called the zero vector and denoted 0 such that
u + 0 = u (additive identity)
5. For every vector u in V , there exists a vector
u + ( u) = 0 (additive inverse)
6. cu 2 V (V is closed under scalar multiplication)
7. c (u + v) = cu + cv
119
u also in V such that
120
CHAPTER 4. VECTOR SPACES
8. (c + d) u = cu + du
9. c (du) = (cd) u
10. 1u = u
Remark 248 When the above is satis…ed, we say that (V; +; :) is a vector space.
So it is important to understand that a vector space is a set of objects together
with two operations acting on these objects.
Remark 249 How the two operations are de…ned depends on the set and the
objects it contains.
Remark 250 The scalars which come into play in scalar multiplication can
be either real numbers, in which case we have a real vector space, or complex
numbers, in which case we have a complex vector space.
Remark 251 Obviously, the vectors we studied in the previous chapter satisfy
these axioms.
Remark 252 From now on, a vector is simply an element of a vector space.
We will still refer to vectors from R2 and R3 to visualize some of their properties.
Remark 253 It can be proven from these axioms that the additive identity is
unique and that for each u, the vector we called u is unique.
To show that a set together with addition and scalar multiplication is a
vector space, we must show that the ten axioms are satis…ed. To show it is not,
it is enough to show one of the axioms is not satis…ed.
Example 254 (2D and 3D vector spaces) R2 and R3 with addition and scalar
multiplication as de…ned in the previous chapter are vector spaces as we saw in
the previous chapter.
Example 255 (n-dimensional vector space) Rn with n
1 is a vector
space. In this case, addition and scalar multiplication are de…ned in a way
similar to that of R2 or R3 .
Example 256 (vector space of 2 2 matrices) Let M22 denote the set of
2 2 matrices. If addition and scalar multiplication are de…ned to be the standard
matrix operations, then M22 is a vector space with these operations. We have
already proven in previous chapters that these operations satis…ed axioms 25 and 7-10. We only need to verify closure under both operations. If A =
a11 a12
b11 b21
and B =
are two arbitrary 2 2 matrices, then
a21 a22
b12 b22
a11 + b11 a12 + b12
A+B =
which is also a 2 2 matrix. And if c is any
a21 + b21 a22 + b22
ca11 ca12
which is also a 2 2 matrix.
scalar, then cA =
ca21 ca22
4.1. VECTOR SPACES
121
Example 257 (vector space of m n matrices) More generally, Mmn , the
set of m n matrices is also a vector space with the standard matrix addition
and scalar multiplication.
Example 258 (vector space of polynomials of degree at most n) For n
0, let Pn denote the set of polynomials of degree at most n. It consists of polynomials of the form p (t) = a0 + a1 t + a2 t2 + ::: + an tn where the constants
a0 :::an and the variable t are real numbers. We de…ne addition of polynomials by (p + q) (t) = (a0 + bb ) + (a1 + b1 ) t + (a2 + b2 ) t2 + ::: + (an + bn ) tn and
(cp) (t) = ca0 + ca1 t + ca2 t2 + ::: + can tn if p (t) = a0 + a1 t + a2 t2 + ::: + an tn
and q (t) = b0 + b1 t + b2 t2 + ::: + bn tn . It is easy to see that closure under both
operations is satis…ed. p + q is indeed a polynomial of degree at most n, so is
cp. The other axioms are satis…ed as they follow from similar properties of real
numbers. The zero polynomial is a polynomial for which all coe¢ cients are 0.
Example 259 (vector space of functions de…ned on some domain) Let
F (D) denote the set of real valued functions de…ned on a set D (typically, D
is R or some interval of the real line). If f and g are two functions in this set,
then we de…ne f + g by (g + g) (x) = f (x) + g (x) and cf by (cf ) (x) = cf (x).
Recall that two functions are equal if and only if their values are equal for every
x 2 D. The zero function is the function whose values are identically 0, that is
f (x) = 0 8x 2 D. It can be shown (see problems) that this is a vector space.
Example 260 (a set that is not a vector space) Let V = fA 2 M22 : jAj =
6 0g
with the standard matrix addition and multiplication. This is not a vector
space, it is not closed under matrix addition. Both I and I are in V . Yet
I + ( I) 2
= V.
4.1.1
Problems
1. Do # 1-18 on pages 226-228.
2. Prove that the set V = f0g with operations de…ned by 0 + 0 = 0 and
c0 = 0 for every scalar c is a vector space.
3. Any plane P in space which goes through the origin has equation ax +
by + cz = 0 where a, b, c are constants. Prove that such a plane is a vector
space.