CONTENTS
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I – Definition and examples
II – Linear independence
III – Rank of vectors
IV – Basic and Dimension
V – Subspaces
I. The Definition and Examples
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A nonempty set V Two operations
Addition Multiplication by scalar
Eight axioms
1. u + v = v + u; 2. (u + v) + w = u + (v + w)
3. There is a zero vector 0 in V such that u + 0 = u
4. For each u in V, there is a vector –u in V such that u + (-u) = 0
VECTOR SPACE V
5. For all scalars a and b and any vector x: (ab)x = a(bx).
6. c(u + v) = cu + cv
7. (c + d)u = cu + du 8. 1u = u
I. The Definition and Examples
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Properties of Vector Space
For each u in V and scalar c:
1)There is a single zero element in an arbitrary vector space
2) There is a single additive inverse vector for every vector u
3) 0u = 0
4) c0 = 0
5) -u = (-1)u
I. The Definition and Examples
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Example 1
V1 ( x1 , x2 , x3 ) xi R
x y ( x1 , x2 , x3 ) ( y1 , y2 , y3 ) ( x1 y1 , x2 y2 , x3 y3 )
x ( x1 , x2 , x3 ) (x1 ,x2 ,x3 )
x1 y1
x y x2 y 2
x y
3 3
V1 - VECTOR SPACE R3
I. The Definition and Examples
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Example 2
V2 ax 2 bx c a, b, c R
p ( x) q ( x) (a1 x 2 b1 x c1 ) (a2 x 2 b2 x c2 )
(a1 a2 ) x 2 (b1 b2 ) x (c1 c2 )
p( x) (ax 2 bx c) ax 2 bx c
a1 a2
p1 ( x) p2 ( x) b1 b2
c c
1 2
V2 - VECTOR SPACE P2 [ x]
I. The Definition and Examples
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Example 3
a b
V3 a , b, c , d R
c d
a1 b1 a2 b2 a1 a2 b1 b2
A1 A2
c d c d c c
1 1 2 2 1 2 1 2 d d
a b a b
A
c d c d
V3 - VECTOR SPACE M 2 [ R ]
I. The Definition and Examples
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Example 4
V4 ( x1, x2 , x3 ) xi R 2 x1 x2 x3 0
The addition and multiplication vector by scalar are the
same as in Example 1.
V4 - VECTOR SPACE
I. The Definition and Examples
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Example 4
V4 ( x1 , x2 , x3 ) xi R x1 x2 2 x3 1
The addition and multiplication vector by scalar are the
same as in Example 1.
V4 - NOT VECTOR SPACE
x (1,2,1) V4 , y (2,3,2) V4
x y (3,5,3) V4
Note: We can define another two operations addition and
multiplication in V1, ( or V2, or V3 ) so that V1 ( or V2, or V3 )
is a vector space.
II. Linear Independence
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V- vector space over K Subset
M {x1 , x2 ,..., xm }
1 , 2 , , m K not all zero
M – linear dependent
1x1 2 x2 m xm 0
1x1 2 x2 m xm 0
1 2 m 0 M – linear independent
II. Linear Independence
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V- vector space over K Subset
M {x1 , x2 ,..., xm }
Vector x in V is called a linear combination of M, if
1 , 2 , , m K
x 1 x1 2 x2 m xm
II. Linear Independence
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Example 5
M { (1, 1, 1 ) ; ( 2 , 1, 3 ) , (1, 2 , 0 ) }
1. Decide whether M is linearly dependent or independent?
2. Is a vector x = (2,-1,3) a linear combination of M?
Solution 1. Suppose ( 1,1,1) ( 2,1, 3 ) ( 1, 2, 0 ) 0
( 2 , 2 , 3 ) ( 0, 0, 0 )
2 0 1 2 1
2 0 A 1 1 2 r( A ) 2
3 0 1 3 0
The system has many solutions, then M is a linearly dependent.
II. Linear Independence
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2. Suppose ( 1,1,1) ( 2,1, 3) ( 1, 2, 0 ) x
( 2 , 2 , 3 ) ( 2, 1, 3)
2 2 1 2 1 2
2 1 (A | b) 1 1 2 1
3 3 1 3 0 3
r(A | b) r(A)
The system is inconsistent. vô nghim
Thus vector x is not a linear combination of M.
II. Linear Independence
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M {x1 , x2 ,, xm }
1x1 2 x2 m xm 0 Homogeneous
system AX=0
Unique trivial
solution X = 0 M – linear independent
nghim duy nht
Non trivial
solution M – linear dependent
vô nghim
II. Linear Independence
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M {x1 , x2 ,, xm }
1 x1 2 x2 m xm x System
AX= b
Consistent
x is a linear combination
of M
Inconsistent x is not a linear
combination of M.
II. Linear Independence
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tp con
Example
Let M { x , y , 2 x 3 y , z } be a subset of a vector
space V.
a. Is a vector 2x + 3y a linear combination of {x, y, z}?
b. Is the set M linear dependent or linear independent?
II. Linear Independence
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Example
Let M = { x, y, z} be a linear independent set in a vector space V.
Show that the set M1= {x+y+2z, 2x+3y+z, 3x+4y+z}
is a linear independent.
Suppose ( x y 2 z ) (2 x 3 y z ) (3 x 4 y z ) 0
( 2 3 ) x ( 3 4 ) y (2 ) z 0
Because of M is a linear independent set, we obtain
2 3 0
0
3 4 0
2 0
Thus M1 is a linear independent set.
II. Linear Independence
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Example
Let M { x , y } be an independent subset of a vector space V.
Decide whether each subset is linearly dependent or linearly
independent.
a. M1 {2x, 3y}
b. M 2 {x+y,2x+3y}
c. M3 {x+y,2x+3y,x-y}
II. Linear Independence
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If M contains a zero vector, then M is linear dependent
M {x1 , x2 ,, xm } - linear dependent
xi - linear combination of the remaining
vectors in M
II. Linear Independence
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If M is a linear dependent, then the set formed from M by
adding some another vectors still linear dependent.
If M is a linear independent, then the set formed from M by
removing some another vectors still linear independent.
Let M be a set containing m vectors: M {x1 , x2 ,..., xm }
Let N be a set containing n vectors: N { y1 , y2 ,..., yn }
If every vector yk from N is a linear combination of M
and n > m, then N is dependent set.
II. Linear Independence
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Example
Let M = { x, y} is a subset of a vector space V.
Is the set M1 ={2x+y, x+3y, 3x+y} a linear dependent or not?
Suppose (2 x y ) ( x 3 y ) (3x y ) 0
(2 3 ) x ( 3 ) y 0
2 3 0
It’s incorrect because M is maybe
3 0 not linear independent
Correct solution. It’s easy to show that every vector from M1 is a
linear combination of M
And the number of vectors in M1 is greater than the number of
vectors in M
Hence M1 is a linear dependent.
II. Linear Independence
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Example
Let {x,y} be a linearly independent subset of a vector space
V and a vector z is not a linear combination of {x ,y}.
Show that the subset {x , y , z } is a linearly independent.
II. Linear Independence
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Example 7
Determine whether the following set of vectors is a linear
dependent or linear independent.
M { (1, 1, 1 ) ; ( 2 , 1, 3 ) , (1, 2 , 0 ) }
II. Linear Independence
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Example 8
Determine whether the following set of polynomials is a
linear dependent or linear independent.
M {x 2 x 1, 2 x 2 3 x 2, 2 x 1}
II. Linear Independence
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Example 9
Determine whether the following set of matrices is a linear
dependent or linear independent.
1 1 2 1 3 4 1 3
M { ; 1 ; ; 1 }
1 0 1 0 1 2
II. Linear Independence
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Example 10
Determine all value(s) m, such that the following set is linear
dependent
M {(1,1,0);(1, 2,1);(m,0,1)}
III. Rank of a set of vectors
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Definition of a rank of a set of vectors
M {x1 , x2 ,, xm ,} V
The rank of M is k0 if there exists k0 independent vectors
from M and any subset of M containing greater than k0
vectors is dependent.
The rank of M is the maximum number of independent
vectors from M.
III. Rank of a set of vectors
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Example 11
Find a rank of following set of vectors
M {(1,1,1,0);(1, 2,1,1);(2,3, 2,1),(1,3,1, 2)}
III. Rank of a set of vectors
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Example
Let M { x , y } be a linearly independent subset of a
vector space V.
Calculate a rank of each following subset.
a. M1 {2x, 3y}
b. M 2 {x,y, 2x 3y}
c. M3 {x,y, 2x 3y, 0}
III. Rank of a set of vectors
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1 2 1 1
A 3 1 0 5
2 4 1 6
The set of row vectors of A.
M {x1 (1, 2,1, 1); x2 (3,1,0,5); x3 (2, 4,1,6)}
The set of column vectors of A.
1 2 1 1
N 3 , 1 , 0 , 5
2 4 1 6
III. Rank of a set of vectors
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Theorem of a Rank
Let A be a mxn matrix over K.
A rank of A is equal to a rank of the set of column
vectors of A.
A rank of A is equal to a rank of the set of row vectors of
A.
III. Rank of a set of vectors
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Example 11
Find a rank of following set of vectors
M {(1,1,1,0);(1,1, 1,1);(2,3,1,1),(3, 4,0, 2)}
Solution
1 1 0 1
1 1 1 1
A
2 3 1 1
3 4 0 2
M is a set of row vectors of A. It follows that rank of M is
equal to a r(A).
III. Rank of a set of vectors
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Let M be a set containing m vectors.
1. If rank of M is equal to m (the number of vectors in M)
then M is independent.
2. If rank of M is less than to m (the number of vectors in
M) then M is dependent.
3. If rank of M is equal to a rank of M adjoint vector x then
x is a linear combination of M.
IV. Basic and Dimension
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Definition of a spanning set
M {x1 , x2 , , xm ,} V
A set M is called a spanning set for vector space V if
any vector from V is a linear combination of M.
M spans V
Vector space V spanned by M
IV. Basic and Dimension
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Example 12
Determine whether a following set is a spanning set for R3.
M {(1,1,1);(1, 2,1);(2,3,1)}
x (x 1 , x 2 , x 3 ) R 3 .
x (x 1 , x 2 , x 3 ) 1 (1,1,1) 2 (1, 2,1) 3 (2, 3,1)
1 2 2 3 x1
Direct calculation shows that
1 2 2 3 3 x2
the system is consistent
1 2 3 x3
Thus, x is a linear combination of M and M is a spanning set for R3
IV. Basic and Dimension
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Example
Determine whether a following set is a spanning set for R3.
M {(1,1, 1);(2,3,1);(3, 4,0)}
x (x 1 , x 2 , x 3 ) R 3 .
x (x 1 , x 2 , x 3 ) 1 (1,1, 1) 2 (2, 3,1) 3 (3, 4, 0)
1 2 2 3 3 x1
1 3 2 4 3 x2
x3
1 2
For x (1, 2,1), the system is inconsistent
The element (1,2,1) is not linear combination of M. Thus M is not a
spanning set for R3.
IV. Basic and Dimension
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Example 13
Determine whether a following set is a spanning set for P2[x].
M {x 2 x 1; 2 x 2 3 x 1; x 2 2 x}
p (x ) ax 2 bx c P2 [x].
2 2 2
p (x ) 1 (x x 1) 2 (2x 3x 1) 3 (x 2x )
1 2 2 3 a
1 3 2 2 3 b
c
1 2
2
For the element p 0 2x x the system has no solution. Thus
the set M is not a spanning set for P2[x].
IV. Basic and Dimension
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M {x1 , x2 ,, xm ,} V
M- independent Span M = V
M- Basic of V
M is a basic
V – finite dimentional
M- finite set Dim V = number of vectors in a
basic of V
If V is not spanned by a finite set, then V is said to be
infinite - dimensional
III. Basic and Dimension
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dim(V) =n
Any set in V containing more than n vectors must be
linearly dependent
Any set in V containing less than n vectors doesn’t span V
Any independent set in V containing exactly n vectors
must be basic of V
Any spanning set of V containing exactly n vectors must
be basic of V
III. Basic and Dimension
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Let S {v1 , v2 ,..., v p } be a set in V and let H = Span {v1 , v2 ,..., v p }
a. If S is a linearly dependent, then the set formed from S by
removing one vector still spans H.
b. If S is a linearly Independent, then any proper subset of S
doesn’t spans H.
IV. Basic and Dimension
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Example 14
Determine whether a following set is a basic for R3.
M {(1,1,1);(2,3,1);(3,1,0)}
IV. Basic and Dimension
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Example 15
Determine whether a following set is a basic for P2[x].
M {x 2 x 1; 2 x 2 x 1; x 2 2 x 2}
