Transcript Document
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 Spanb1 , , 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 ifv1 , 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 Spanv1 , , 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
4
2
0
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
6
4
0
1
0
2
2
1
1
4
0
1
Theorem
The pivot columns of a matrix A form a basis for Col A.