The document discusses vector spaces and related concepts: 1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials. 2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations. 3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.

1. Prepared by
Tanuj Parikh

2. Chapter Two: Vector Spaces
Vector space ~ Linear combinations of vectors.
I. Definition of Vector Space
II. Linear Independence
III. Basis and Dimension
• Topic: Fields
• Topic: Crystals
• Topic: Voting Paradoxes
• Topic: Dimensional Analysis
Ref: T.M.Apostol, “Linear Algebra”, Chap 3, Wiley (97)

3. I. Definition of Vector Space
I.1. Definition and Examples
I.2. Subspaces and Spanning Sets

4. Algebraic Structures
Ref: Y.Choquet-Bruhat, et al, “Analysis, Manifolds & Physics”, Pt
I., North Holland (82)
Structure Internal Operations Scalar Multiplication
Group * No
Ring, Field * , + No
Module / Vector Space + Yes
Algebra + , * Yes
Field = Ring with idenity & all elements except 0 have inverses.
Vector space = Module over Field.

5. I.1. Definition and Examples
Definition 1.1: (Real) Vector Space ( V, § ;  )
A vector space (over ) consists of a set V along with 2 operations ‘§’ and ‘¨’ s.t.
(1) For the vector addition § :
" v, w, u Î V
a) v § w Î V ( Closure )
b) v § w = w § v ( Commutativity )
c) ( v § w ) § u = v § ( w § u ) ( Associativity )
d) $ 0 Î V s.t. v § 0 = v ( Zero element )
e) $ -v Î V s.t. v § (-v) = 0 ( Inverse )
(2) For the scalar multiplication ¨ :
" v, w Î V and a, b Î , [  is the real number field (,+,´)
f) a ¨ v Î V ( Closure )
g) ( a + b ) ¨ v = ( a ¨ v ) § (b ¨ v ) ( Distributivity )
h) a ¨ ( v § w ) = ( a ¨ v ) § ( a ¨ w )
i) ( a ´ b ) ¨ v = a ¨ ( b ¨ v ) ( Associativity )
j) 1 ¨ v = v
 § is always written as + so that one writes v + w instead of v § w
 ´ and ¨ are often omitted so that one writes a b v instead of ( a ´ b ) ¨ v

6. Definition in Conventional Notations
Definition 1.1: (Real) Vector Space ( V, + ;  )
A vector space (over ) consists of a set V along with 2 operations ‘+’ and ‘ ’ s.t.
(1) For the vector addition + :
" v, w, u Î V
a) v + w Î V ( Closure )
b) v + w = w + v ( Commutativity )
c) ( v + w ) + u = v + ( w + u ) ( Associativity )
d) $ 0 Î V s.t. v + 0 = v ( Zero element )
e) $ -v Î V s.t. v -v = 0 ( Inverse )
(2) For the scalar multiplication :
" v, w Î V and a, b Î , [  is the real number field (,+,´) ]
a) a v Î V ( Closure )
b) ( a + b ) v = a v + b v ( Distributivity )
c) a ( v + w ) = a v + a w
d) ( a ´ b ) v = a ( b v ) = a b v ( Associativity )
e) 1 v = v

7. Example 1.3: 2
x y
2 is a vector space if 1 1
æ ö æ ö
ax by
ax by
æ + ö
= ç ¸ è + ø
x y 1 1
a b a b
+ = ç ¸+ ç ¸
x y
è 2 ø è 2
ø
2 2
" a,bÎR
0
0
= æ ö ç ¸
è ø
with 0
Proof it yourself / see Hefferon, p.81.
Example 1.4: Plane in 3.
The plane through the origin 0
ì æ x
ö ü
= ï ç ¸ í ç ¸ + + = ï ý ï î çè ø¸ ï þ
P y x y z
z
is a vector space.
P is a subspace of 3.
Proof it yourself / see Hefferon,
p.82.

8. Example 1.5:
Let § & ¨ be the (column) matrix addition & scalar multiplication, resp., then
( n, + ;  ) is a vector space.
( n, + ;  ) is not a vector space since closure is violated under scalar multiplication.
Example 1.6:
0
0
0
0
V
ì æ ö ü
ï ç ¸ ï = ï ç ¸ ï í ç ¸ ý ï ç ¸ ï îï è ø ïþ
Let then (V, + ;  ) is a vector space.
Definition 1.7: A one-element vector space is a trivial space.

9. Example 1.8: Space of Real Polynomials of Degree n or less, n
ìï ïü = í Î ý
îï ïþ
P å R { 2 3 }
0
n
k
n k k
k
a x a
=
3 0 1 2 3 k P = a + a x + a x + a x a ÎR
n is a vector space with vectors
a º å
n n
The kth component of a is
a + b = å +å ( ) k k k a + b = a + b
Vector
addition: 0 0
k k
a x b x
k k
k k
= =
Scalar multiplication:
æ ö
a å
b b a x
= ç ¸
è 0
ø
n
k
k
k
=
Zero element:
= å +
= å
0 = å ( ) 0 k i.e., 0 = "k
0
0
n
k
k
x
=
0
n
k
k
k
a x
=
( )
0
n
k
k k
k
a b x
=
0
n
k
k
k
ba x
=
i.e.
,
( ) k k i.e. b a = ba
,
E.g.
,
-a º å - ( ) k k i.e. -a = -a
å ÎP L ÎR
is isomorphic to n+1 with ( ) 1
n 0
0
~ , ,
n
k n
k n n
k
a x a a +
=
Inverse: ( )
0
n
k
k
k
a x
=
,
( ) k k a = a

10. Example 1.9: Function Space
The set { f | f :  →  } of all real valued functions of natural numbers is
a vector space if
Vector addition: ( f1 + f2 ) ( n) º f1 ( n) + f2 ( n)
Scalar multiplication: ( a f ) ( n) º a f ( n)
nÎN
aÎR
Zero
element:
zero(n) = 0
Inverse: ( - f ) (n) º - f ( n)
f ( n ) is a vector of countably infinite dimensions: f = ( f(0), f(1), f(2), f(3), … )
E.g.,
f ( n) = n2 +1 ~ f = (1, 2, 5,10,L)

11. Example 1.10: Space of All Real Polynomials, 
ìï ïü = í Î Î ý
îï ïþ
P å R N
0
,
n
k
k k
k
a x a n
=
 is a vector space of countably infinite dimensions.
å ÎP ( L ) ÎR
0 1 2
0
k ~ , , ,
k
k
a x a a a
¥
¥
=
Example 1.11: Function Space
The set { f | f :  →  } of all real valued functions of real numbers is a
vector space of uncountably infinite dimensions.

12. Example 13: Solution Space of a Linear Homogeneous Differential Equation
ì ü
= í ® + = ý
î þ
2
2 S f : d f f 0
R R is a vector space with
d x
Vector addition: ( f + g) ( x) º f ( x) + g ( x)
Scalar multiplication: ( a f ) ( x) º a f ( x)
Zero
element:
zero(x) = 0
Inverse: ( - f ) (x) º - f ( x)
Closure:
aÎR
+ = + = ( ) ( )
2 2
2 2 d f f 0 & d g g 0
d x d x
2
d a f bg
a f bg
2 0
d x
+
→ + + =
Example 14: Solution Space of a System of Linear Homogeneous Equations

13. Remarks:
• Definition of a mathematical structure is not unique.
• The accepted version is time-tested to be most concise & elegant.
Lemma 1.16: Lose Ends
In any vector space V,
1. 0 v = 0 .
2. ( -1 ) v + v = 0 .
3. a 0 = 0 .
" v ÎV and a Î .
Proof:
1. 0 = v - v = (1+ 0) v - v = v + 0v - v = 0v
2.
( -1) v + v = ( -1+1) v = 0v = 0
3.
a 0 = a ( 0 v) = ( a 0) v = 0 v = 0

14. Exercises 2.I.1.
1. At this point “the same” is only an intuition, but nonetheless for
each vector space identify the k for which the space is “the same” as k.
(a) The 2´3 matrices under the usual operations
(b) The n ´ m matrices (under their usual operations)
(c) This set of 2 ´ 2 matrices
ìï æ a
0
ö ïü í ç ¸ a + b + c
= 0
ý
îï è b c
ø ïþ
2.
(a) Prove that every point, line, or plane thru the origin in 3 is a
vector space under the inherited operations.
(b) What if it doesn’t contain the origin?

15. I.2. Subspaces and Spanning Sets
Definition 2.1: Subspaces
For any vector space, a subspace is a subset that is itself a vector space,
under the inherited operations.
Note: A subset of a vector space is a subspace iff it is closed under § & ¨.
ì æ x
ö ü
= ï ç ¸ í ç ¸ + + = ï ý ï î çè ø¸ ï þ
Example 2.2: Plane in 3 P y x y z
0
z
is a subspace of 3.
→ It must contain 0. (c.f. Lemma 2.9.)
Proof: Let ( ) ( ) 1 1 1 1 2 2 2 2 , , , , , T T r = x y z r = x y z ÎP
→ 1 1 1 2 2 2 x + y + z = 0 , x + y + z = 0
( ) 1 2 1 2 1 2 1 2 , , T ar + br = ax + bx ay + by az + bz
with ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 1 1 2 2 2 ax + bx + ay + by + az + bz = a x + y + z + b x + y + z
→ 1 2 ar + br ÎP " a,bÎR QED
= 0

16. Example 2.3: The x-axis in n is a subspace.
( ,0, ,0) -axis T Proof follows directly from the fact that r = x L Î x
Example 2.4:
• { 0 } is a trivial subspace of n.
• n is a subspace of n.
Both are improper subspaces.
All other subspaces are proper.
Example 2.5: Subspace is only defined wrt inherited operations.
({1},§ ; ) is a vector space if we define 1§1 = 1 and a¨1=1 "aÎ.
However, neither ({1},§ ; ) nor ({1},+ ; ) is a subspace of the vector space
(,+ ; ).

17. Example 2.6: Polynomial Spaces.
n is a proper subspace of m if n < m.
Example 2.7: Solution Spaces.
The solution space of any real linear homogeneous ordinary
differential equation,  f = 0,
is a subspace of the function space of 1 variable { f :  →  }.
Example 2.8: Violation of Closure.
+ is not a subspace of  since (-1) v Ï + " vÎ +.

18. Lemma 2.9:
Let S be a non-empty subset of a vector space ( V, + ;  ).
W.r.t. the inherited operations, the following statements are equivalent:
1. S is a subspace of V.
2. S is closed under all linear combinations of pairs of vectors.
3. S is closed under arbitrary linear combinations.
Proof: See Hefferon, p.93.
Remark: Vector space = Collection of linear combinations of vectors.

19. Example 2.11: Parametrization of a Plane in 3
ì æ x
ö ü
= ï ç ¸ í ç ¸ - 2 + = 0
ï ý ï î çè ø¸ ï þ
S y x y z
z
is a 2-D subspace of 3.
ì æ 2
y - z
ö ü
= ï ç ¸ í ç y ¸ y ,
z
Î R
ï ý ï î çè z
ø¸ ï þ
ì æ 2 ö æ - 1
ö ü
= ï ç í y 1 ¸+ ç ¸ z ç 0 ¸ ç ¸ y ,
z
Î R
ï ý ï î çè 0 ø¸ èç 1
ø¸ ï þ
i.e., S is the set of all linear combinations of 2 vectors (2,1,0)T, &
(-1,0,1)T.
Example 2.12: Parametrization of a Matrix Subspace.
ìï æ a
0
ö ïü = í ç ¸ + + = 0
ý
îï è ø ïþ
L a b c
b c
is a subspace of the space of 2´2 matrices.
ìï æ- b - c
0
= ö í ç ¸ b ,
c
Î R
ïü b c
ý
îï è ø ïþ
ìï æ - 1 0 ö æ - 1 0
= b ¸+ c ö í ç ç ¸ b ,
c
Î R
ïü 1 0 0 1
ý
îï è ø è ø ïþ

20. Definition 2.13: Span
Let S = { s1 , …, sn | sk Î ( V,+, ) } be a set of n vectors in vector space V.
The span of S is the set of all linear combinations of the vectors in S, i.e.,
ì ü
= í Î Î ý
î þ
å s s R with span Æ = { 0 }
span S c S c
1
,
n
k k k k
k
=
Lemma 2.15: The span of any subset of a vector space is a subspace.
Proof:
n n
Let S = { s, …, s| sÎ ( V,+,) }
1 n k  and u =å s v =å s Î
u ,
v span S
k k k k
= =
k k
1 1
w = u + v =å + s
® a b ( au bv
)
1
n
k k k
k
=
=å s Î " a,bÎR
1
n
k k
k
w span S
=
QED
Converse: Any vector subspace is the span of a subset of its members.
Also: span S is the smallest vector space containing all members of
S.

21. Example 2.16:
For any vÎV, span{v} = { a v | a Î } is a 1-D subspace.
Example 2.17:
Proof:
The problem is tantamount to showing that for all x, y Î, $ unique a,b Î s.t.
1 1
1 1
x
æ ö = a æ ö + b
æ ö ç è y
¸ ç ¸ ç ¸ ø è ø è - ø
i.e., a b x
+ =
- =
a b y
has a unique solution for arbitrary x & y.
Since 1 ( )
a = x + y 1 ( )
2
b = x - y " x, yÎR QED
2
2 1 1
,
1 1
span
ì æ ö æ ö ü í ç ¸ ç ¸ ý = î è ø è - ø þ
R

22. Example 2.18: 2
Let S = span{ 3x - x2 , 2x } = { a ( 3x - x2 ) + 2bx a,bÎR }
Question:
0 2 0
?
c S = =P
Answer is yes since
ì ü
= í ý
î þ
å = subspace of 2 ?
1 c = 3a + 2b 2 c = -a
1 3
2
a = -c b = ( c - a
) 2 1
and
1 3
2
( ) 1 2
= c + c
2
1
k
k
k
c x
=
Lesson: A vector space can be spanned by different sets of vectors.
Definition: Completeness
A subset S of a vector space V is complete if span S = V.

23. Example 2.19: All Possible Subspaces of 3
See next section for proof.
Planes thru
0
Lines thru 0

24. Exercises 2.I.2
ì æ x
ö ü
ï ç í ç y ¸ ï çè ø¸ ¸ x + y + z
= ý ï z
ï î þ
1. Consider the set 1
under these operations.
x x x x 1
y y y y
z z z z
æ ö æ + - ç 1 ¸+ ç 2 ö æ 1 2
ö
¸= ç ¸ çç 1 ¸¸ çç + 2 ¸¸ çç 1 2
¸¸ è 1 ø è + 2 ø è 1 2
ø
x rx r 1
r y r y
z rz
æ ö æ - + ö
ç ¸= ç ¸ çç ¸¸ çç ¸¸ è ø è ø
(a) Show that it is not a subspace of 3. (Hint. See Example 2.5).
(b) Show that it is a vector space.
( To save time, you need only prove axioms (d) & (j), and closure
under all linear combinations of 2 vectors.)
(c) Show that any subspace of 3 must pass thru the origin, and so any
subspace of 3 must involve zero in its description.
Does the converse hold?
Does any subset of 3 that contains the origin become a subspace
when given the inherited operations?

25. 2. Because ‘span of’ is an operation on sets we naturally consider how it
interacts with the usual set operations. Let [S] º Span S.
(a) If S Í T are subsets of a vector space, is [S] Í [T] ?
Always? Sometimes? Never?
(b) If S, T are subsets of a vector space, is [ S È T ] = [S] È [T] ?
(c) If S, T are subsets of a vector space, is [ S Ç T ] = [S] Ç [T] ?
(d) Is the span of the complement equal to the complement of the span?
3. Find a structure that is closed under linear combinations, and yet is not
a vector space. (Remark. This is a bit of a trick question.)