Toán học - Chapter 4: Vector space

Math Dept, Faculty of Applied Science, HCM University of Technology ------------------------------------------------------------------------------------- Linear Algebra Chapter 4: VECTOR SPACE • Instructor Dr. Dang Van Vinh (12/2006) CONTENTS --------------------------------------------------------------------------------------------------------------------------- I – Definition and examples V – Subspaces II – Linear independence IV – Basic and Dimension III – Rank of vectors VECTOR SPACE V I. The Definition and Examples --------------------------------------------------------------------------------------------------------------------------- 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 1. u + v = v + u; 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 A nonempty set V Two operations Multiplication by scalar Addition Eight axioms I. The Definition and Examples --------------------------------------------------------------------------------------------------------------------------- For each u in V and scalar c: 3) 0u = 0 4) c0 = 0 5) -u = (-1)u Properties of Vector Space 1)There is a single zero element in an arbitrary vector space 2) There is a single additive inverse vector for every vector u I. The Definition and Examples ---------------------------------------------------------------------------------------------------------------------------  RxxxxV i  ),,( 3211 ),,(),,(),,( 332211321321 yxyxyxyyyxxxyx  ),,(),,( 321321 xxxxxxx            33 22 11 yx yx yx yx Example 1 V1 - VECTOR SPACE 3R I. The Definition and Examples ---------------------------------------------------------------------------------------------------------------------------  RcbacbxaxV  ,,22 Example 2 )()()( )()()()( 2121 2 21 22 2 211 2 1 ccxbbxaa cxbxacxbxaxqxp   cbxaxcbxaxxp   22 )()(          21 21 21 21 )()( cc bb aa xpxp V2 - VECTOR SPACE ][2 xP I. The Definition and Examples ---------------------------------------------------------------------------------------------------------------------------              Rdcba dc ba V ,,,3 Example 3 V3 - VECTOR SPACE ][2 RM                      2121 2121 22 22 11 11 21 ddcc bbaa dc ba dc ba AA                dc ba dc ba A    I. The Definition and Examples --------------------------------------------------------------------------------------------------------------------------- 4 1 2 3 1 2 3( , , ) 2 0iV x x x x R x x x      The addition and multiplication vector by scalar are the same as in Example 1. V4 - VECTOR SPACE Example 4 I. The Definition and Examples ---------------------------------------------------------------------------------------------------------------------------  12),,( 3213214  xxxRxxxxV i The addition and multiplication vector by scalar are the same as in Example 1. V4 - NOT VECTOR SPACE 44 )2,3,2(,)1,2,1( VyVx  4)3,5,3( Vyx  Example 4 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 --------------------------------------------------------------------------------------------------------------------------- V- vector space over K 1 2{ , ,..., }mM x x x Subset M – linear dependent 1 2, , , m K    not all zero 1 1 2 2 0m mx x x      M – linear independent 1 1 2 2 0m mx x x      1 2 0m      II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- V- vector space over K 1 2{ , ,..., }mM x x x Subset 1 2, , , m K    1 1 2 2 m mx x x x      Vector x in V is called a linear combination of M, if II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- { (1,1,1 ) ; ( 2 ,1, 3 ) , (1, 2 , 0 ) }M  Example 5 1. Decide whether M is linearly dependent or independent? 2. Is a vector x = (2,-1,3) a linear combination of M? Solution 1. 2 2 3 0 0 0( , , ) ( , , )              2 0 2 0 3 0                    1 2 1 1 1 2 1 3 0 A           2r( A )  The system has many solutions, then M is a linearly dependent. Suppose 1 1 1 2 1 3 1 2 0 0( , , ) ( , , ) ( , , )     II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 2. Suppose 1 1 1 2 1 3 1 2 0( , , ) ( , , ) ( , , ) x     2 2 3 2 1 3( , , ) ( , , )               2 2 2 1 3 3                     1 2 1 2 1 1 2 1 1 3 0 3 (A | b)            r(A | b) r(A) Thus vector x is not a linear combination of M. The system is inconsistent. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , }mM x x x  1 1 2 2 0m mx x x      Homogeneous system AX=0 Unique trivial solution X = 0 M – linear dependent Non trivial solution M – linear independent II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , }mM x x x  1 1 2 2      m mx x x x System AX= b Consistent x is not a linear combination of M. Inconsistent x is a linear combination of M Let be a subset of a vector space V. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- { , , 2 3 , } M x y x y z Example 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 --------------------------------------------------------------------------------------------------------------------------- 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 ( 2 ) (2 3 ) (3 4 ) 0x y z x y z x y z           ( 2 3 ) ( 3 4 ) (2 ) 0x y z                  Because of M is a linear independent set, we obtain 2 3 0 3 4 0 2 0                     0      Thus M1 is a linear independent set. Let be an independent subset of a vector space V. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- { , }M x y Example a. b. 1 2 3M { x, y} 2 M {x+y,2x+3y} c. 3 M {x+y,2x+3y,x-y} Decide whether each subset is linearly dependent or linearly independent. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , }mM x x x  - linear dependent - linear combination of the remaining vectors in M ix If M contains a zero vector, then M is linear dependent  II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 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: 1 2{ , ,..., }mM x x x Let N be a set containing n vectors: 1 2{ , ,..., }nN y y y If every vector yk from N is a linear combination of M and n > m, then N is dependent set. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 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 ) ( 3 ) (3 ) 0x y x y x y        (2 3 ) ( 3 ) 0x y            It’s incorrect because M is maybe not linear independent 2 3 0 3 0               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 --------------------------------------------------------------------------------------------------------------------------- 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 is a linearly independent. { , , }x y z II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- { (1,1,1) ; ( 2 ,1, 3 ) , (1, 2 , 0 ) }M  Example 7 Determine whether the following set of vectors is a linear dependent or linear independent. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- Example 8 Determine whether the following set of polynomials is a linear dependent or linear independent. 2 2{ 1,2 3 2,2 1}M x x x x x      II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- 1 1 2 1 3 4 1 3 { ; ; ; } 1 0 1 1 0 1 1 2 M                          Example 9 Determine whether the following set of matrices is a linear dependent or linear independent. II. Linear Independence --------------------------------------------------------------------------------------------------------------------------- Example 10 Determine all value(s) m, such that the following set is linear dependent {(1,1,0);(1,2,1);( ,0,1)}M m III. Rank of a set of vectors --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , , }mM x x x V   Definition of a rank of a set of vectors 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 --------------------------------------------------------------------------------------------------------------------------- Example 11 Find a rank of following set of vectors {(1,1,1,0);(1,2,1,1);(2,3,2,1),(1,3,1,2)}M  Let be a linearly independent subset of a vector space V. III. Rank of a set of vectors --------------------------------------------------------------------------------------------------------------------------- { , }M x y Example a. b. 1 2 3M { x, y} 2 2 3M {x,y, x y}  Calculate a rank of each following subset. c. 3 2 3 0M {x,y, x y, }  III. Rank of a set of vectors --------------------------------------------------------------------------------------------------------------------------- 1 2 1 1 3 1 0 5 2 4 1 6 A          1 2 3{ (1,2,1, 1); (3,1,0,5); ( 2,4,1,6)}M x x x      The set of row vectors of A. The set of column vectors of A. 1 2 1 1 3 , 1 , 0 , 5 2 4 1 6 N                                             III. Rank of a set of vectors --------------------------------------------------------------------------------------------------------------------------- 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 --------------------------------------------------------------------------------------------------------------------------- Example 11 Find a rank of following set of vectors {(1,1,1,0);(1,1, 1,1);(2,3,1,1),(3,4,0,2)}M   Solution 1 1 1 0 1 1 1 1 2 3 1 1 3 4 0 2 A             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 --------------------------------------------------------------------------------------------------------------------------- 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 --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , , }mM x x x V   Definition of a spanning set 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 --------------------------------------------------------------------------------------------------------------------------- Example 12 Determine whether a following set is a spanning set for R3. {(1,1,1);(1,2,1);(2,3,1)}M  1 2 3 3( , , ) .  x x x x R Thus, x is a linear combination of M and M is a spanning set for R3 1 2 3 1 2 31 1 1 1 2 1 2 3 1( , , ) ( , , ) ( , , ) ( , , )     x x x x 1 2 3 1 1 2 3 2 1 2 3 3 2 2 3                      x x x Direct calculation shows that the system is consistent IV. Basic and Dimension --------------------------------------------------------------------------------------------------------------------------- Example Determine whether a following set is a spanning set for R3. {(1,1, 1);(2,3,1);(3,4,0)}M   1 2 3 3( , , ) .  x x x x R 1 2 3 1 2 31 1 1 2 3 1 3 4 0( , , ) ( , , ) ( , , ) ( , , )      x x x x 1 2 3 1 1 2 3 2 1 2 3 2 3 3 4                     x x x For , the system is inconsistent 1 2 1( , , )x The element (1,2,1) is not linear combination of M. Thus M is not a spanning set for R3. IV. Basic and Dimension --------------------------------------------------------------------------------------------------------------------------- Example 13 Determine whether a following set is a spanning set for P2[x]. 2 2 2{ 1;2 3 1; 2 }M x x x x x x      2 2( ) [x].    p x ax bx c P 2 2 2 1 2 31 2 3 1 2( ) ( ) ( ) ( )         p x x x x x x x 1 2 3 1 2 3 1 2 2 3 2                    a b c For the element the system has no solution. Thus 2 0 2 p x x the set M is not a spanning set for P2[x]. IV. Basic and Dimension --------------------------------------------------------------------------------------------------------------------------- 1 2{ , , , , }mM x x x V   M- independent Span M = V M- Basic of V M is a basic M- finite set V – finite dimentional 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 --------------------------------------------------------------------------------------------------------------------------- dim(V) =n Any set in V containing less than n vectors doesn’t span V  Any set in V containing more than n vectors must be linearly dependent  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 --------------------------------------------------------------------------------------------------------------------------- Let be a set in V and let H = Span 1 2{ , ,..., }pS v v v 1 2{ , ,..., }pv v v 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 --------------------------------------------------------------------------------------------------------------------------- Example 14 Determine whether a following set is a basic for R3. {(1,1,1);(2,3,1);(3,1,0)}M  IV. Basic and Dimension --------------------------------------------------------------------------------------------------------------------------- Example 15 Determine whether a following set is a basic for P2[x]. 2 2 2{ 1;2 1; 2 2}M x x x x x x      