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4 Vector Spaces
4.3 Linearly Independent Sets; Bases
REVIEW
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
Definition
REVIEW
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 )
REVIEW
Theorem 1
If v1 , , v p are in a vector space V,
then Span v1 ,  , v p is a subspace of V.


REVIEW
Definition
The null space of an m n matrix A, written as Nul A, is
the set of all solutions to Ax=0.

Nul A  x | Ax  0 and x  R n

Theorem 2
The null space of an m n matrix A is a subspace of  n .
REVIEW
Definition
The column space of an m n matrix A, written as Col A, is
the set of all linear combinations of the columns of A.
Col A  Span{a1 ,, an }
Note: Col A  {b | Ax  b for some x  n }
Theorem 3
m

The column space of an m n matrix A is a subspace of
.
REVIEW
Definition:
A linear transformation T from a vector space V into a vector
space W is a rule assigns to each vector x in V a unique vector
T(x) in W, such that
(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:
(ii) T(cu)=cT(u) for all u and all scalars c.
kernel of T  x | T ( x)  0 and x V 
Range of T  {b W | T ( x)  b for some x V }
If T(x)=Ax for some matrix A,
Kernel of T = Nul A
Range of T = Col A.
4.3 Linearly Independent Sets; Bases
Purpose: To study the vectors that span
a vector space (or a subspace) as
efficiently as possible.
Linear Independence

v , v ,, v 
1
2
p
in V is linearly independent
x1 v1  x2 v2    x p v p  0 has only the trivial solution.

x1 v1  x2 v2    x p v p  0  x1  x2    x p  0
Tips to determine the linear dependence
A set is linearly dependent, if it satisfies one of the following:
1. A set has two vectors and one is a multiple of the other.
2. A set has two or more vectors and one of the vectors is a
linear combination of the others.
3. A set contains more vectors than the number of entries in
each vector.
4. A set contains the zero vector.
Theorem 4


An indexed set v1 , v2 , , v p of two or more vectors, with v1  0,
is linearly dependent
if and only if some v j ( j  1) is a linear combination of the
preceding vectors,
v1 , v2 ,  , v j 1.
Example: p1 (t )  3, p2 (t )  2t , p3 (t )  t  1
Is p1 , p2 , p3 linearly dependent?
Definition
Let H be a subspace of a vector space V.
An indexed set of vectors   b1 ,, b p  in V is a basis for H if
i)  is a linearly independent set, and
ii) the subspace spanned by  coincides with H; i.e.
H  Spanb1 , , b p 
Examples:
1. Let A be an invertible n n matrix. Then the columns of A
form a basis for  n. Why?
2. Let e1 , , en be the columns of the n n identity matrix I.
Then, e1 ,, en  is called the standard basis for  n .
1
 2
3
4. Let v1  2, v2  0, v3  0
 
 
 
3
1
1
.
Determine if v1 , v2 , v3  is a basis for  3 .
1
 2
  2
0 
5. Let v1  2, v2  0, v3   3 , v4  2.
 
 
 
 
3
1
  1
1
Determine ifv1 , v2 , v3 , v4 is a basis for  3 .
  4
  2
6
6. Let v   6 , v   0 , v   9.
1
  2   3  
 2 
 1 
 3
Determine if v1 , v2 , v3  is a basis for  3 .
The Spanning Set Theorem
Let S  v1 ,  , v p  be a set in V, and let H  Spanv1 , , v p .
a. If one of the vectors in S, say vk , is a linear combination
of the remaining vectors in S, then the set formed from S by
removing vk still spans H.
b. If H  0, some subset of S is a basis for H.
Example: Find a basis for Col A, where
1
0
A
0

0
3
0
0
0
4
2
0
0
3
4 
 1  2
0
0 

0
1 
Example: Find a basis for Col B, where
1
 3
B
2

1
4
1
1
2
2
6
4
2
0
4
 1 0 
0
1

1  1
Theorem
The pivot columns of a matrix A form a basis for Col A.