西安建筑科技大学：《线性代数》课程双语教学课件_线性方程组解的结构

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

Linear dependence and independence VI Since a;=(x1i, X2i, ., Xni),then c1a1+c2a2+…+cmm=0. It equal to the system of equations C1x1+c2×2+……+cmX1m=0 C1x21+c2x2+…+cm×m=0 c1xn+c2Xn+…+cmXm=0 Let x=(c1, C2,..., Cm), then the matrix equation is Ax=0 From the theorem, a linear equation Ax=0 has nontrivial if and only if rca<m

Linear dependence and independence VI Proof. Since αi = (x1i , x2i , · · · , xni) T , then c1α1 + c2α2 + · · · + cmαm = 0. It equal to the system of equations    c1x11 + c2x12 + · · · + cmx1m = 0 c1x21 + c2x22 + · · · + cmx2m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c1xn1 + c2xn2 + · · · + cmxnm = 0 . Let x = (c1, c2, · · · , cm), then the matrix equation is Ax = 0. From the theorem, a linear equation Ax = 0 has nontrivial if and only if R(A) < m. () May 3, 2006 6 / 40

Linear dependence and independence VI Corollary Let a1, 02, .. am be m vectors in R, if m>n, then al, .. am be linearly dependent. Let a be a matrix of a1, 02, .. am, then a is an m x n(or n x m) matrix. Since R(A<min(m, n)=n< m, so the vectors be linearly dependent. Example: Vectors a1=(1, 1, 2), a2=(0, 1,0), a3=(0, 0, 1)be linearly ependen

Linear dependence and independence VII Corollary Let α1, α2, · · · , αm be m vectors in Rn , if m > n, then α1, · · · , αm be linearly dependent. Proof. Let A be a matrix of α1, α2, · · · , αm, then A is an m × n(or n × m) matrix. Since R(A) < min(m, n) = n < m, so the vectors be linearly dependent. Example: Vectors α1 = (1, 1, 2)T , α2 = (0, 1, 0)T , α3 = (0, 0, 1)T be linearly dependent. () May 3, 2006 7 / 40

Linear dependence and independence I By the proof of Theorem 4.ll, we can prove the following properties o a set consists of one vector a is linear independent if and only if a+0 o If a collection A of vectors a1, 02, .. an are represented by a collection b of vectors B1, B2,.. Bm and n> m, then A is linearly dependent o If an vector B is linearly represented by al, a2, .. an, then the set an, B is linearly dependent o Suppose that the collection of vectors a1, 02, .. an is linearly independent, but the collection a1,a2,……,an,β is linear dependence,thenβcanb linearly represented by a1, 02, ..,an in a unique way o If the collection of vectors a1, 02, .. ar is linearly dependent, then 01, 02,., ar, ar+1,., am is also linearly dependent o If the collection of vectors a1, 02, .. am is linearly independent, then a1,a2,.,a(r< m)is also linearly independent

Linear dependence and independence I By the proof of Theorem 4.11, we can prove the following properties: A set consists of one vector α is linear independent if and only if α 6= 0. If a collection A of vectors α1, α2, · · · , αn are represented by a collection B of vectors β1, β2, · · · , βm and n > m, then A is linearly dependent. If an vector β is linearly represented by α1, α2, · · · , αn, then the set α1, α2, · · · , αn, β is linearly dependent. Suppose that the collection of vectors α1, α2, · · · , αn is linearly independent, but the collection α1, α2, · · · , αn, β is linear dependence, then β can be linearly represented by α1, α2, · · · , αn in a unique way. If the collection of vectors α1, α2, · · · , αr is linearly dependent, then α1, α2, · · · , αr , αr+1, · · · , αm is also linearly dependent. If the collection of vectors α1, α2, · · · , αm is linearly independent, then α1, α2, · · · , αr(r < m) is also linearly independent. () May 3, 2006 8 / 40

Linear dependence and independence I Exercise: Suppose the collection of vectors a1, a2, a3 is linearly dependent the collection of vectors a2, a3, a4 is linearly independent, proof 1. Vector a1 can be linearly represented by a2, a3 2. Vector aa can not be linearly represented by a1, a2, a3

Linear dependence and independence I Exercise: Suppose the collection of vectors α1, α2, α3 is linearly dependent, the collection of vectors α2, α3, α4 is linearly independent, proof 1. Vector α1 can be linearly represented by α2, α3. 2. Vector α4 can not be linearly represented by α1, α2, α3. () May 3, 2006 9 / 40