Toán học - Chapter 4: Vector space (cont)

Math Dept, Faculty of Applied Science, HCM University of Technology ------------------------------------------------------------------------------------- Linear Algebra Chapter 4: VECTOR SPACE (cont.) • Instructor Dr. Dang Van Vinh (12/2006) CONTENTS --------------------------------------------------------------------------------------------------------------------------- I – Coordinates of a vector II – Subspaces III – The intersection and sum of subspaces I. Coordinates of a vector ------------------------------------------------------------------------------------------------- Let E ={e1, e2, , en} be an ordered basic for a vector space V and let x be an element of a vector space V. Definition of coordinates of a vector 1 1 2 2 ... n nx x e x e x e    The coefficients x1, x2, , xn in the expansion are called the coordinates of x with respect to E, and from now on, [x]E will denote the column vector 1 2[ ]E n x x x x              I. Coordinates of a vector ------------------------------------------------------------------------------------------------- 2 2 2 2Let { 1; 2 1; 2} be a basic of [ ].E x x x x x x P x       Example Find the vector p(x), whose coordinates with respect to E are 3 [ ( )] 5 2 Ep x            I. Coordinates of a vector ------------------------------------------------------------------------------------------------- 3Let {(1,1,1);(1,0,1);(1,1,0)} be a basic of .E R Example Let x = (3,1,-2) be an element of R3. Find the coordinates of V with respect to the basic E. I. Coordinates of a vector ------------------------------------------------------------------------------------------------- 2 2Let { 1; 1;2 1} be a basic of [ ].E x x x x P x     Example Find the coordinates of vector f(x) = 3x2+4x-1 with respect to E. I. Coordinates of a vector ------------------------------------------------------------------------------------------------- 1 1 2 22. [ ]E n n x y x y x y x y              1 1 2 21. n n x y x y x y x y           1 2[ ]E n y y y y              Properties of coordinates of a vectors 1 2[ ]E n x x x x              1 23. [ ]E n x x x x                  I. Coordinates of a vector ------------------------------------------------------------------------------------------------- 2 2 2 2Let { 1;3 2 1;2 } be a subset of [ ].M x x x x x x P x      Example Determine whether M is a linearly dependent or linearly independent subset. Subset F II. Subspaces --------------------------------------------------------------------------------------------------------------------------- V- vector space Subset F 2 operations in V Vector Space F pace F II. Subspaces --------------------------------------------------------------------------------------------------------------------------- A nonempty subset F of a vector space V is a subspace of V if and only if the following statements are true. 1. , : f g F f g F    2. , for any scalar : f F f F    Theorem 1 II. Subspaces ---------------------------------------------------------------------------------------------------------------------------  1 2 3 3 1 2 3( , , ) | 2 0F x x x R x x x     Example 1. Show that F is a subspace of R3 2. Find a dimension and basic of F. II. Subspaces ---------------------------------------------------------------------------------------------------------------------------  3( ) | (1) 0 & (0) 0F p x R p p    Example 1. Show that F is a subspace of R3. 2. Find a dimension and basic of F. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Example  2 1 1 [ ] | 0 2 2 F A M R A           1. Show that F is a subspace of M2[R] 2. Find a dimension and basic of F. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Span{M}=Span 1 2 1 1 2 2{ , ,..., } { }n n n iv v v v v v R         1 2{ , , , }nM v v v V  1. Span{M} is a subspace 2. The dim(Span{M}) is equal to the rank of M. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Let (1,1,1);(2,1,1);(3,1,1)F   Find the basic and dimension of F. Example II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Let 2 2 21,2 3 1, 2 2F x x x x x x        Find the basic and dimension of F. Example II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Example 2 where , 2 a b a b F a b R b a            Find a dimension and basic of F. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Example 1 1 2 1 3 1 1 0 , , , 2 1 0 1 2 1 2 0 F                          Find a dimension and basic of F. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Let (1, 2,3); {(1,1,1);(2,1,0);(3, 1,3)}x M    Is x in the subspace spanned by M? Example II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Let (1,0, ); {(1,1,1);(2,3,1);(3,2,0)}x m M  For what value(s) of m will x be in the subspace spanned by M? Example 3. The product of one solution by a scalar will be a solution. 2. The sum of two solutions will be a solution. 1. The set E0 is not an empty set. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Let AX = 0 be a homogeneous system. Let E0 be a set of all solutions of the system. E0 is a subspace of R n and is called Nullspace of A. II. Subspaces --------------------------------------------------------------------------------------------------------------------------- Subspace Span of some set Solutions of AX = 0 Nul A = { | 0}nx R Ax  i j m nA a  The null space Null A of an mxn matrix A is a subspace of Rn. III. Intersection and Sum of subspaces --------------------------------------------------------------------------------------------------------------------------- Dim(E0) = number of unknowns – r(A). Theorem Let AX = 0 be a homogeneous system. Example Find the basic and dimension of nullspace of following matrix 1 1 2 1 1 2 3 4 3 1 3 2 5 4 3 1 1 1 1 2 A            III. Intersection and Sum of subspaces --------------------------------------------------------------------------------------------------------------------------- Let F and G be two subspaces of vector space V. The intersection of two subspaces F and G is a subset of V, denoted by Definition of Intersection of two subspaces { | and }F G x V x F x G    The Sum of two subspaces F and G is a subset of V, denoted by Definition of Sum of two subspaces { | where and }F G f g f F g G     III. Intersection and Sum of subspaces --------------------------------------------------------------------------------------------------------------------------- 2. Theorem 1. are two subspaces of V. andF G F G dim( ) dim( ) dim( ) dim( )F G F G F G     Remark F G F F G V    F G G F G V    III. Intersection and Sum of subspaces --------------------------------------------------------------------------------------------------------------------------- Steps to finding subspace F+G 1. Find the spanning set of F. Suppose this set is {f1, f2, , fn} 1 2 1 23. , ,..., , , ,...,n nF G f f f g g g   2. Find the spanning set of G. Suppose this set is {g1, g2, , gn} II. Intersection and Sum of two subspaces --------------------------------------------------------------------------------------------------------------------------- Let F and G be two subspaces of R3, where Example  1 2 3 1 2 3( , , ) | 2 0}F x x x x x x     1 2 3 1 2 3( , , ) | 0}G x x x x x x    1. Find the dimension and basic of .F G 2. Find the dimension and basic of .F G II. Intersection and Sum of two subspaces --------------------------------------------------------------------------------------------------------------------------- Let F and G be two subspaces of R3, where Example  1 2 3 1 2 3( , , ) | 0}F x x x x x x    (1,01,);(2,3,1)G   1. Find the dimension and basic of .F G 2. Find the dimension and basic of .F G II. Intersection and Sum of two subspaces --------------------------------------------------------------------------------------------------------------------------- Let F and G be two subspaces of R4, where Example  1 2 3 41 2 3 4 1 2 3 4 0 ( , , , ) 2 2 0 x x x x F x x x x x x x x             1. Find the dimension and basic of .F G 2. Find the dimension and basic of .F G  1 2 3 41 2 3 4 1 2 3 4 0 ( , , , ) 3 2 2 3 0 x x x x G x x x x x x x x             II. Intersection and Sum of two subspaces --------------------------------------------------------------------------------------------------------------------------- .F G Let F and G be two subspaces of R3, where Example (1,0,1);(1,1,1) F   1. Find the dimension and basic of 2. Find the dimension and basic of .F G (1,1,0);(2,1,1) G   II. Intersection and Sum of two subspaces --------------------------------------------------------------------------------------------------------------------------- .F G Let F and G be two subspaces of P2[x], where Example 2{ ( ) [ ] | (1) 0}F p x P x p   1. Find the dimension and basic of 2. Find the dimension and basic of .F G 2{ ( ) [ ] | ( 1) 0}G p x P x p   