Linear dependence and independence I 4.2 Linear dependence and independence Linear space are generalization of vector spaces, so basic concepts in vector spaces may be defined in linear space Definition The vectors vi, v2, .. Vn in a vector space v are said to be linear dependence if there are numbers c1, c2, .. cr not all zero such that C1v+c2v+…+cnv=0. Otherwise,Ⅵ,v2,……, Vn are said to be linear independent Example: 1. Let x=(1, 2, 3) then the vectors e1, e2, e3, x are linearly dependent. Since e1+2e2+3e3-x=0 In this case G= 1, C2=2, c3=3 1
Linear dependence and independence I 4.2 Linear dependence and independence Linear space are generalization of vector spaces, so basic concepts in vector spaces may be defined in linear space. Definition The vectors v1, v2, · · · , vn in a vector space V are said to be linear dependence if there are numbers c1, c2, · · · , cr not all zero such that c1v1 + c2v2 + · · · + cnvn = 0. Otherwise, v1, v2, · · · , vn are said to be linear independent. Example: 1. Let x = (1, 2, 3)T , then the vectors e1, e2, e3, x are linearly dependent. Since e1 + 2e2 + 3e3 − x = 0. In this case, c1 = 1, c2 = 2, c3 = 3, c4 = −1. () May 3, 2006 1 / 40
Linear dependence and independence Il 2. If x and y are linearly dependent in R2, then CIx+Cry=0 where Cl and c2 are not both 0. Suppose C1#0, we can write o That is to say, if two vectors in R2 are linearly dependent, one of the vectors can be written as a scalar multiple of the other. o Thus, if V1, v2, .. vn are linearly dependent, then at least one of the vectors can be expressed as a linear combination of the rest of the vectors
Linear dependence and independence II 2. If x and y are linearly dependent in R 2 , then c1x + c2y = 0 where c1 and c2 are not both 0. Suppose c1 6= 0, we can write x = − c2 c1 y. That is to say, if two vectors in R 2 are linearly dependent, one of the vectors can be written as a scalar multiple of the other. Thus, if v1, v2, · · · , vn are linearly dependent, then at least one of the vectors can be expressed as a linear combination of the rest of the vectors. () May 3, 2006 2 / 40
Linear dependence and independence Ill 3. The vectors(1,)and(1, 2) are linearly independent, since if a(1)+(2)-(8) then C1+C2=0 +2c2=0 and the only solution to this system is c1=0, c2=0. Thus,fⅵ,v,……, n are linearly endent, then no vector can be expressed as a linear combination of the rest vectors. Equivalently c1Ⅵ+c2v+…+Cnvn=Q implies that C1=0=C2=
Linear dependence and independence III 3. The vectors (1, 1)T and (1, 2)T are linearly independent, since if c1 1 1 + c2 1 2 = 0 0 then c1 + c2 = 0 c1 + 2c2 = 0 and the only solution to this system is c1 = 0, c2 = 0. Thus, if v1, v2, · · · , vn are linearly independent, then no vector can be expressed as a linear combination of the rest vectors. Equivalently c1v1 + c2v2 + · · · + cnvn = 0 implies that c1 = 0 = c2 = · · · = cn. () May 3, 2006 3 / 40
Linear dependence and independence IV Determine the vector collection a1=(1, 1, 1), 02=(0, 2, 5),a3=(1, 3, 6) are linearly independent in R3? tion a1(1,1,1)7+c2(0,2,5)7+c3(1,3,6)7=(0,0.0)7, 0 C1+2c2+3c3=0 C1+5c2+6c3 The coefficient matrix of this system 101 101 A=(a1,a2,a3) 156 055
Linear dependence and independence IV Example Determine the vector collection α1 = (1, 1, 1)T , α2 = (0, 2, 5)T , α3 = (1, 3, 6)T are linearly independent in R 3 ? Solution: If c1(1, 1, 1)T + c2(0, 2, 5)T + c3(1, 3, 6)T = (0, 0, 0)T , then c1 + c3 = 0 c1 + 2c2 + 3c3 = 0 c1 + 5c2 + 6c3 = 0. The coefficient matrix of this system A = (α1, α2, α3) = 1 0 1 1 2 3 1 5 6 r2−r1 r3−r1 −→ 1 0 1 0 2 2 0 5 5 r3− 5 2 r2 −→ 1 0 1 0 2 2 0 0 0 () May 3, 2006 4 / 40
Linear dependence and independence V since R(A)=2<3(the number of the vectors), so the system has nontrivial solution and the vectors are linearly dependent This illustrates a special case of the following theorem Theorem Let a1, 02, .. am be vectors in R with a;=(x1i, X2i, . .. Xni)for i=1,2,……,n, and let A=(xi),j=1,2,…,n. that is,thea;' s form the columns of A. Then the vectors a1.a am will be linearly dependent if and only if R(A)< m
Linear dependence and independence V since R(A) = 2 < 3(the number of the vectors), so the system has nontrivial solution and the vectors are linearly dependent. This illustrates a special case of the following theorem. Theorem Let α1, α2, · · · , αm be vectors in Rn with αi = (x1i , x2i , · · · , xni) T for i = 1, 2, · · · , n, and let A = (xji), j = 1, 2, · · · , n. that is, the αi ’s form the columns of A. Then the vectors α1, α2, · · · , αm will be linearly dependent if and only if R(A) < m. () May 3, 2006 5 / 40