MAE101 ALG chapter 3 the vector space rn

ĐẠI HỌC FPT CẦN THƠ Chapter The vector space Rn Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Contents 5.1 Subspaces and Spanning sets 5.2 Independence and Dimension 5.3 Orthogonality 5.4 Rank of a Matrix Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Subspace of Rn Definition of subspace of • •• Let Ø≠U be a subset of Rn • • U is called a subspace of Rn if:  S1 The zero vector is in U vector zero vector  S2 If X,Y are in U then X+Y is in U  S3 If X is in U then aX is in U for all real number a Ex1 U={(a,a,0)|aR} is a subspace of R3 n  the zero vector of R , (0,0,0)U aX • • •  (a,a,0), (b,b,0)U(a,a,0)+(b,b,0)=(a+b,a+b,0)U • • • X+Y U•  If (a,a,0) U and k R, then k(a,a,0)=(ka,ka,0)U • Ex2 U={(a,b,1): a,b R} is not a subspace of R3 Y X  (0,0,0)U  U is not a subspace Ex3 U={(a,|a|,0)|a R} is not a subspace of R3  (-1,|-1|,0), (1,|1|,0)U but (0,2,0) U  U is not a subspace Rn ▪ ▪ ▪ ▪ ▪ • Dr Tran Quoc Duy U Mathematics for Engineering n ĐẠI HỌC FPT CẦN THƠ V={[0 a 0]T in 3: a Z} ▪ U={[a 3a]T in 3: aR} ▪ W={[5a b a-b]T in 3: a,bR} ▪ Q={[a b |a+b|]T: a } ▪ H={[a b ab]T: a,b } ▪ ▪ Nhận xét: trường hợp sau không không gian vector ▪ có thành phần khác khơng ▪ có hệ số bậc cao tích ▪ có dấu | | ▪ có a a+1 chẳng hạn P={(x,y,z)| x-2y+z=0 and 2x-y+3z=0} P is called the solution space of the system x-2y+z=0 and 2x-y+3z=0 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Note ⚫ A subspace either has only one or infinite many vectors ⚫ Example, {0} has only vector ⚫ If a subspace U has nonzero vector X then aX is also in U (by S3) Then U has infinite many vector Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Null space and image space of a matrix ▪ ▪ ▪ A is an mxn matrix, if X is nx1 matrix then AX is mx1 matrix nullA = {X in Rn: AX=0} A imA = {AX: X is in Rn} nullA m • imA n zero vector nullA ={X Rn:AX=0} is a subspace of Rn:  A.0=00nullA  X,Y nullA AX=0, AY=0 A(X+Y)=AX+AY=0 (X+Y) nullA  X nullA, a R  AX=0  A(aX)=a(AX)=0  aXnullA Dr Tran Quoc Duy imA ={AX:X  Rn}is a subspace of Rm:  0=A.00imA  AX,AY imA AX+AY=A(X+Y)=AZ AX+AY imA  AX imA, a R  a(AX)=A(aX)=AZ  a(AX)imA Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Null space null A={X:AX=0} −1   ▪ For example, A = 2 1   23  x   x    x    x − 0           nullA =   y  : A  y  =    =   y  :  y =     0        0    z   z    z    z           x    t  x − y =     =  y  :  = t :t    x + y + z = 0    z      −5t       Dr Tran Quoc Duy      Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Eigenspaces (không gian riêng) ▪ ▪ ▪ Suppose A is an nxn matrix and λ is an eigenvalue of A Eλ(A)={X: AX=λX} is an subspace of Rn For example, x+3  −3 −1 A=  c x = det xI − A = = ( x + 3)( x − ) ( ) ( ) A  x −   c A ( x ) =  x = −3  x = 0  0  t  x = −3 :  →  X = (or X= ( t ,0 ))      0  0 −5  0 0  5   t  x = 2:  X =   −5t  0     E−3 =  X : AX = −3 X  = ( t ,0 ) : t  E2 =  X : AX = X  = ( t , −5t ) : t  Dr Tran Quoc Duy   Các không gian riêng ứng với GTR Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Spanning sets (hệ sinh) ▪ ▪ ▪ ▪ ▪ ▪ ▪ Y=k1X1+k2X2+…+knXn is called a linear combination of the vectors X1,X2,…,Xn The set of all linear combinations of the the vectors X1,X2,…,Xn is called the span of these vectors, denoted by span{X1,X2,…,Xn } This means, span{X1,X2,…,Xn} = {k1X1+k2X2+…+knXn :kiR is arbitrary} span{X1,X2,…,Xn} is a subspace of Rn For example, span{(1,0,1),(0,1,1)}={a(1,0,1)+b(0,1,1) :a,bR} And we have (1,2,3)span{(1,0,1),(0,1,1)} because (1,2,3)= 1(1,0,1)+ 2(0,1,1) (2,3,2)span{(1,0,1),(0,1,1)} because (2,3,2)≠a(1,0,1)+b(0,1,1) for all a,b ▪ Nếu U=span{X,Y} ta nói U KG sinh {X,Y} hay hệ {X,Y} sinh KG U Khi đó, U chứa tất vector có dạng aX+bY với a, b số thực tùy ý ▪ vector Zspan{X,Y} có số thực a,b cho Z=aX+bY hay hệ pt Z=aX+bY có nghiệm a,b ▪Ta nói Z tổ hợp tuyến tính (linear combination) X,Y Z=aX+bY hay Zspan{X,Y} Dr Tran Quoc Duy Mathematics for Engineering Examples ĐẠI HỌC FPT CẦN THƠ ▪ If x=(1,3,-5) is expressed as a linear combination of the vectors v1 = (1, 1, 1); v2 =(1,1,-1); v3 = (1, 0, 2); then the coefficient of v3 is: A B C -2  D E ▪ x is expressed as a linear combination of v1, v2, v3 means x=av1+bv2+cv3 for some a,b,c and c is called the coefficient of v3 ▪ the system is 1 1 1 1 a+b+c = 1 1  a =1 -2 -6  b =2 0 -1 a+b = 1 a – b +2c =-5 -1 -5  c =-2 0 -1 -2 -6 ▪ Which of the vectors below is a linear combination of u=(1,1,2); v=(2,3,5)? A (0,1,1)  B (1,1,0) C (1,1,1) D (1,0,1)  E (0,0,1) ▪ Có thể giải biến đổi sơ cấp ma trận chứa vector cột sau: u v A B C D E u v A B C D E u v A B C D E  1  -2 1  -2 1 1 1 0  0 -1 0  0 -1 1 1 1 -2 -1 -1 0 -2 -1 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Orthogonal Set (hệ trực giao) A set {x1,x2,…,xm} is called orthogonal set if xi is not zero vector and xi•xj=0 for all i≠j ▪ For example, {(1,-1);(1,1)} is an orthogonal set in R2 ▪ {(1,1,1);(-1,0,1);(0,1,0)} is not orthogonal set but {(1,0,1);(0,1,0)} is a orthogonal set ▪ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Orthogonal Set (hệ trực giao) A orthogonal set {xi} is called orthonormal set (hệ trực chuẩn) is xi is unit vector for all i For example, {(1,0,0);(0,1,0)} is orthonormal ▪ {(-3,0,4);(4,5,3)} is a orthogonal set, not a orthonormal set However, the set ▪ 1   is orthonormal  ( −3, 0, ) ; ( 4,5,3) 5  Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Examples ▪ The standard basis of Rn {E1,E2,…,En} is orthonormal ▪ If {F1,F2,…,Fk} is orthogonal then {a1F1,a2F2,…,akFk} is also orthogonal for any nonzero scalar ▪ Every orthogonal set is a linearly independent set Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Pythagoras’s Theorem ▪ If {F1,F2,…,Fk} is orthogonal then F1 + F2 + + Fk Dr Tran Quoc Duy 2 = F1 + F2 + + Fk Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Expansion Theorem ▪ Let {F1,F2,…,Fk} be a orthogonal basis of a subspace U and X is in U Then X= X • F1 F1 Dr Tran Quoc Duy F1 + X • F2 F2 F2 + + X • Fn Fn Fk Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 5.4 Rank of a matrix Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Rank of a matrix ▪ ▪ ▪ If A is carried to row-echelon form then rankA=number of leading 1’s If A is an mxn matrix then rankA≤min{n,m} rankA=rank(AT) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ rowA and colA subspaces ▪ ▪ ▪ ▪ rowA=span{rows of matrix A} colA=span{columns of A} dim(rowA)=dim(colA)=rankA For example, find bases of colA and rowA if  1 −1  3 5   A =  −2 −3 −4    1 −    −1  Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ  1 −1  1 −1    0 −1 −1     A =  −2 −3 −4  → 0 −1      1 − 0      −1  0 −1  1 −1  1 −1  0 −1 −1 0 −1 −1     → 0 −2  → 0 −2      0 0  0 0  0 −3  0 0  A basis of rowA is {r1,r2,r3,r4} and dim(rowA)=4 A basis of colA is {c1,c2,c3,c4} and dim(colA)=4 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem ▪ ▪ ▪ An nxn matrix A is invertible if and only if rankA=n If an mxn matrix B has rank n then the n columns of B is linearly independent If A is mxn matrix and m>n then the set of m rows of A is not independent Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem ▪ ▪ ▪ ▪ ▪ If an mxn matrix A has rank r then The equation AX=0 has n-r basic solutions X1,X2,…,Xn-r {X1,X2,…,Xn-r} is a basis of nullA Dim nullA=n-r imA=colA and Dim imA=dim colA=rankA=r Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Examples ▪ ▪ ▪ ▪ ▪ Find a basis and dimU if: U=span{(1,-1,2,0);(-2,1,0,1);(-1,0,0,1); (1,0,1,2)} U=span{(1,-1,3,0);(5,-2,4,3);(-2,0,7,1)} U=span{(-1,4,3);(3,0,-2);(-6,2,0)} U={(a,b,c):a+b+c=0} U={[a b 0]T: a,b in R} Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Thanks Dr Tran Quoc Duy Mathematics for Engineering ... of subspace of • •• Let Ø≠U be a subset of Rn • • U is called a subspace of Rn if:  S1 The zero vector is in U vector zero vector  S2 If X,Y are in U then X+Y is in U  S3 If X is in U then... Y=k1X1+k2X2+…+knXn is called a linear combination of the vectors X1,X2,…,Xn The set of all linear combinations of the the vectors X1,X2,…,Xn is called the span of these vectors, denoted by span{X1,X2,…,Xn }... subspace U has nonzero vector X then aX is also in U (by S3) Then U has infinite many vector Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Null space and image space of a matrix