Section 4.1: Vector Spaces and Subspaces
REVIEW
Recall the following algebraic properties of
R
n
n
For all u, v, w in R and all scalars c and d :
i) u  v  v  u
v) c(u  v )  cu  cv
ii) (u  v)  w  u  ( v  w )
vi) (c  d)u  cu  du
iii) u  0  0  u  u
vii) c(du)  (cd)u
iv) u  (-u)  -u  u  0 viii) 1  u  u
( -u  (-1)u )
Definition
A vector space is a nonempty set V of objects, called vectors,
on which are defined two operations, called addition and
multiplication by scalars, subject to the ten axioms:
For all u, v, w in V and all scalars c and d,
1) u  v  V
2) u  v  v  u
3)(u  v)  w  u  (v  w)
4) There is a zero vector 0 such that u  0  u
5) For each u  U , there is -u such that u  (u )  0
6) cu  V
7)c(u  v)  cu  cv
8)(c  d)u  cu  du
9) c(du )  (cd)u
10)1u  u
Examples
1. n


2. Pn  p(t )  a0  a1t  a2t 2    ant n | ai  , i  0,1,, n
= the set of polynomials of degree at most n
For p(t )  a0  a1t    ant n and q(t )  b0  b1t    bnt n ,
p(t )  q(t ) 
c  p(t ) 
3. FD : the set of all real-valued functions defined on a set D.
4. M n : the set of n  n matrices
Definition
A subspace of a vector space V is a subset H of V that satisfies
a. The zero vector of V is in H.
b. H is closed under vector addition.
(For each u and v  H , u  v  H)
c. H is closed under multiplication by scalars.
(For each u  H and each scalar c, cu  H )
Properties a-c guarantee that a subspace H of V is
itself a vector space.
Why? a, b, and c in the defn are precisely axioms 1,
4, and 6. Axioms 2, 3, 7-10 are true in H because
they apply to all elements in V, including those in H.
Axiom 5 is also true by c.
Thus every subspace is a vector space and conversely,
every vector space is a subspace (or itself or possibly
something larger).
Examples:
1. The zero space {0}, consisting of only the zero vector in V
is a subspace of V.
2. Pn is a subspace of P, the set of all polynomial s with real
coefficien ts.
3. M n : the set of n  n matrices
Dn : the set of n  n diagonal matrices
 Dn is a subspace of M n
4. R 2 is not a subspace of R 3 . Why not?
5. Given v1 and v2 in a vector space V, let
Show that H is a subspace of V.
H  Span{v1, v2}
Theorem 1
If v1 ,, v p are in a vector space V,
then Span v1 ,  , v p is a subspace of V.


Example:
1. Show H  {( a  3b, b  a, a,2b) | a, b  R} is a subspace of R 4 .
2. Is K  {( a  3b, b  a, a  1,2b  2) | a, b  R} a subspace of R 4 ?