Math 28S
Vector Spaces
Fall 2011
Definition: Given a field F , a vector space over F is a set V together with two operations:
• addition: + : V × V → V (i.e. (u, v) 7→ u + v)
• scalar multiplication, F × V → V (i.e. (c, v) 7→ cv)
such that the following rules (called the “Vector Space Laws”) are satisfied:
1. Addition is closed: For all u, v ∈ V , u + v ∈ V .
2. Scalar multiplication is closed: For all c ∈ F and all v ∈ V , cv ∈ V .
3. Addition is commutative: For all u, v ∈ V , u + v = v + u.
4. Addition is associative: For all u, v, w ∈ V , u + (v + w) = (u + v) + w.
5. Additive identity element: There exists an element of V called 0 such that v + 0 = v for all
v ∈V.
6. Additive inverses exist: For all v ∈ V , there exists an element −v ∈ V such that v + (−v) = 0.
7. Distributivity I: For all c ∈ F and all u, v ∈ V , c(u + v) = cu + cv.
8. Distributivity II: For all c, d ∈ F and all v ∈ V , (c + d)v = cv + dv.
9. Scalar multiplication is associative: For all c, d ∈ F and all v ∈ V , (cd)v = c(dv).
10. Identity element for scalar multiplication: 1v = v where 1 refers to the multiplicative identity
element of F .
Language: Given a vector space V , the field F over which it lies is called the underlying field of
V ; elements of the underlying field are called scalars; elements of the vector space are called vectors.
Notation: Vectors are usually referred to by boldface letters (like v) when typed, and as let−
ters with arrows over them (like →
v ) when hand-written. However, sometimes we get lazy and just
→
−
refer to a vector with a letter (like v). The zero scalar is denoted 0; the zero vector is denoted 0 or 0 .
Theorem: R2 = {(x, y) : x, y ∈ R} is a vector space over R, where the addition and scalar
multiplication are defined coordinate-wise, i.e.
(x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 )
and c(x1 , y1 ) = (cx1 , cy1 ).
Proof: Let u, v, and w be arbitrary elements of R2 , and let c, d ∈ R. By definition of R2 , we
have u = (u1 , u2 ); v = (v1 , v2 ) and w = (w1 , w2 ). We check the vector space laws one by one:
1. Addition is closed: This is obvious by the definition of vector addition.
2. Scalar multiplication is closed: This is obvious by the definition of scalar multiplication.
3. Addition is commutative: We need to check u + v = v + u:
u + v = (u1 , u2 ) + (v1 , v2 )
= (u1 + v1 , u2 + v2 )
(by defn of + in R2 )
= (v1 + u1 , v2 + u2 )
(since + is commutative in R)
= (v1 , v2 ) + (u1 , u2 )
(by defn of + in R2 )
= v + u.
4. Addition is associative:
u + (v + w) = (u1 , u2 ) + ((v1 , v2 ) + (w1 , w2 ))
= (u1 , u2 ) + (v1 + w1 , v2 + w2 )
(by defn of + in R2 )
= (u1 + (v1 + w1 ), u2 + (v2 + w2 ))
(by defn of + in R2 )
= ((u1 + v1 ) + w1 , (u2 + v2 ) + w2 )
(since + is associative in R)
= (u1 + v1 , u2 + v2 ) + (w1 , w2 )
= ((u1 , u2 ) + (v1 , v2 )) + (w1 , w2 )
= (u + v) + w.
(by defn of + in R2 )
(by defn of + in R2 )
Math 28S
Vector Spaces
Fall 2011
5. Additive identity element: Let 0 = (0, 0). Then
v + 0 = (v1 , v2 ) + (0, 0)
(by defn of + in R2 )
= (v1 + 0, v2 + 0)
= (v1 , v2 )
(since 0 is additive identity in R)
= v.
6. Additive inverses exist: Let −v = (−v1 , −v2 ). Then
v + (−v) = (v1 , v2 ) + (−v1 , −v2 )
= (v1 + (−v1 ), v2 + (−v2 ))
= (0, 0)
(by defn of + in R2 )
(by properties of additive inverses in R)
= 0 (by defn of 0).
7. Distributivity I:
c(u + v) = c((u1 , u2 ) + (v1 , v2 ))
(by defn of + in R2 )
= c(u1 + v1 , u2 + v2 )
= (c(u1 + v1 ), c(u2 + v2 ))
(by defn of scalar multiplication)
= (cu1 + cv1 , cu2 + cv2 )
(by distributivity of R)
= (cu1 , cu2 ) + (cv1 , cv2 )
(by defn of + in R2 )
= c(u1 , u2 ) + c(v1 , v2 )
(by defn of scalar multiplication)
= cu + cv.
8. Distributivity II: For all c, d ∈ F and all v ∈ V , (c + d)v = cv + dv.
(c + d)v) = (c + d)(v1 , v2 )
= ((c + d)v1 , (c + d)v2 )
(by defn of scalar multiplication)
= (cv1 + dv1 , cv2 + dv2 )
(by distributivity of R)
= (cv1 , cv2 ) + (dv1 , dv2 )
(by defn of + in R2 )
= c(v1 , v2 ) + d(v1 , v2 )
(by defn of scalar multiplication)
= cv + dv.
9. Scalar multiplication is associative: For all c, d ∈ F and all v ∈ V , (cd)v = c(dv).
(cd)v = (cd)(v1 , v2 )
= ((cd)v1 , (cd)v2 )
(by defn of scalar multiplication)
= (c(dv1 ), c(dv2 ))
(by associativity of · in R)
= c(dv1 , dv2 )
(by defn of scalar multiplication)
= c(d(v1 , v2 ))
(by defn of scalar multiplication)
= c(dv).
10. Identity element for scalar multiplication:
1v = 1(v1 , v2 )
= (1v1 , 1v2 )
= (v1 , v2 )
(by defn of scalar multiplication)
(since 1 is mult. identity in R)
= v.
Since all the laws hold, R2 is indeed a vector space over R. Math 28S
Typical Vector Spaces
Fall 2011
1. “Traditional” vectors: Given any field F and any n ∈ N, F n = {(x1 , ..., xn ) : xj ∈ F ∀ j} is a
vector space over F , where the addition and scalar multiplication are defined coordinate-wise.
(The proof of this is the same as the proof that R2 is a vector space, essentially.)
2. Fields: Given any field F , F is a vector space over itself (where the addition and scalar
multiplication are the field operations). In particular, F = F 1 .
3. Zero vector spaces: Given any field F , {0} is a vector space over F . In particular, F 0 = {0}.
4. C is a vector space over R.
5. C is also a vector space over C. But, the vector spaces C (over R) and C (over C) are two
different vector spaces.
6. Matrix spaces: Given any field F , the set of m×n matrices (this means m rows and n columns)
with elements in F , denoted Mmn (F ), is a vector space over F where the addition and scalar
multiplication are defined entry-wise. (Notation: the set of square n × n matrices with entries
in F is denoted Mn (F ) rather than Mnn (F ).)
7. Sequence spaces: In these examples, the addition and scalar multiplication are defined termby-term, i.e.
(x1 , x2 , x3 , ...)+(y1 , y2 , y3 , ...) = (x1 +y1 , x2 +y2 , x3 +y3 , ...)
and c(x1 , x2 , ...) = (cx1 , cx2 , ...).
(a) Given any field F , the set F N of infinite sequences of elements of F is a vector space over
F (where the addition and scalar multiplication are done term-by-term).
(b) The set F ∞ of infinite sequences where all but finitely many elements of the sequence are
0 also forms a vector space over F , with the same operations. Here, some care needs to
be taken to ensure that addition is closed.
(c) The set of convergent sequences of real numbers forms a vector space over R.
8. Function spaces: For all these spaces of functions, the addition is described by (f + g)(x) =
f (x) + g(x) and the scalar multiplication is (cf )(x) = c · f (x)
(a) Polynomials: Given any field F , the set F [x] of polynomials whose coefficients are in F
is a vector space over F , where the addition and scalar multiplication come from usual
addition and scalar multiplication of functions.
(b) Continuous functions: The set C(R, R) of continuous functions from R to R is a vector
space over R.
(c) Differentiable functions: The set of differentiable functions from R to R is a vector space
over R is a vector space over R.
(d) Analytic functions: The set of analytic functions from R to R is a vector space over R is
a vector space over R (a function is analytic if it can be written as a power series which
converges everywhere).
R∞
(e) L2 functions: The set of “measurable” functions from R to R satsfying −∞ |f (x)|2 dx <
∞ is a vector space over R (loosely speaking, a function is measurable if for every a
Rb
and b in R, a f (x) dx exists... all continuous and piecewise-continuous functions are
measurable).