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1、3 3 The Vector Space R The Vector Space Rn n 3.2 Vector space Properties of Rn 3.3 Examples of Subspaces 3.4 Bases for Subspaces 3.5 Dimension 3.6 Orthogonal Bases for SubspacesCore SectionsIn mathematics and the physical sciences,the term vector is applied to a wide variety of objects.Perhaps the m
2、ost familiar application of the term is to quantities,such as force and velocity,that have both magnitude and direction.Such vectors can be represented in two space or in three space as directed line segments or arrows.As we will see in chapter 5,the term vector may also be used to describe objects
3、such as matrices,polynomials,and continuous real-valued functions.3.1 IntroductionIn this section we demonstrate that Rn,the set of n-dimensional vectors,provides a natural bridge between the intuitive and natural concept of a geometric vector and that of an abstract vector in a general vector space
4、.3.2 VECTOR SPACE PROPERTIES OF Rn.numbers real ,:2121nnnxxxxxxXXRThe Definition of Subspaces of RnA subset W of Rn is a subspace of Rn if and only if the following conditions are met:(s1)*The zero vector,is in W.(s2)X+Y is in W whenever X and Y are in W.(s3)aX is in W whenever X is in W and a is an
5、y scalar.Example 1:Let W be the subset of R3 defined by.numbers realany and,:32321321xxxxxxxxXXWVerify that W is a subspace of R3 and give a geometric interpretation of W.Solution:Step 1.An algebraic specification for the subset W is given,and this specification serves as a test for determining whet
6、her a vector in Rn is or is not in W.Step 2.Test the zero vector,of Rn to see whether it satisfies the algebraic specification required to be in W.(This shows that W is nonempty.)Verifying that W is a subspace of RnStep 3.Choose two arbitrary vectors X and Y from W.Thus X and Y are in Rn,and both ve
7、ctors satisfy the algebraic specification of W.Step 4.Test the sum X+Y to see whether it meets the specification of W.Step 5.For an arbitrary scalar,a,test the scalar multiple aX to see whether it meets the specification of W.Example 3:Let W be the subset of R3 defined by.1:21numbers realany x and x
8、 ,21xxXXWShow that W is not a subspace of R3.Example 2:Let W be the subset of R3 defined by.,:321number realany x,3xx,2xx11312xxxXXWVerify that W is a subspace of R3 and give a geometric interpretation of W.Example 4:Let W be the subset of R2 defined by.,:21integersany x and x21xxXXWDemonstrate that
9、 W is not a subspace of R2.Example 5:Let W be the subset of R2 defined by.,:21 0 x or 0 x21eitherwherexxXXWDemonstrate that W is not a subspace of R2.Exercise P175 18 323.3 EXAMPLES OF SUBSPACESIn this section we introduce several important and particularly useful examples of subspaces of Rn.The spa
10、n of a subsetTheorem 3:If v1,vr are vectors in Rn,then the set W consisting of all linear combinations of v1,vr is a subspace of Rn.If S=v1,vr is a subset of Rn,then the subspace W consisting of all linear combinations of v1,vr is called the subspace spanned by S and will be denoted by Sp(S)or Spv1,
11、vr.For example:(1)For a single vector v in Rn,Spv is the subspace Spv=av:a is any real number.(2)If u and v are noncollinear geometric vectors,then Spu,v=au+bv:a,b any real numbers(3)If u,v,w are vectors in R3,and are not on the same space,then Spu,v,w=au+bv+cw:a,b,c any real numbersExample 1:Let u
12、and v be the three-dimensional vectors210 vand 012uDetermine W=Spu,v and give a geometric interpretation of W.The null space of a matrixWe now introduce two subspaces that have particular relevance to the linear system of equations Ax=b,where A is an(mn)matrix.The first of these subspaces is called
13、the null space of A(or the kernel of A)and consists of all solutions of Ax=.Definition 1:Let A be an(m n)matrix.The null space of A denoted N(A)is the set of vectors in Rn defined by N(A)=x:Ax=,x in Rn.Theorem 4:If A is an(m n)matrix,then N(A)is a subspace of Rn.Example 2:Describe N(A),where A is th
14、e(3 4)matrix.142145121311ASolution:N(A)is determined by solving the homogeneous system Ax=.This is accomplished by reducing the augmented matrix A|to echelon form.It is easy to verify that A|is row equivalent to.000000211003201Solving the corresponding reduced system yields x1=-2x3-3x4 x2=-x3+2x4,10
15、23011223243434343xxxxxxxxXWhere x3 and x4 are arbitrary;that is,.numbers realany and ,10230112:)(4343xxxxXXANExample 5:Let S=v1,v2,v3,v4 be a subset of R3,where.1-52 v,541 v,532 v,1214321andvShow that there exists a set T=w1,w2 consisting of two vectors in R3 such that Sp(S)=Sp(T).Solution:let.15515
16、4322121ASet row operation to A and reduce A to the following matrix:000012104501000012102121363012102121.155154322121ASo,Sp(S)=av1+bv2:a,b any real numberBecause Sp(T)=Sp(S),then Sp(T)=av1+bv2:a,b any real numberFor example,we set 7015322121323211vvw31053212122212vvw.155154322121AThe solution on P184,152541532121TAAnd the row vectors of AT are precisely the vectors v1T,v2T,v3T,and v4T.It is straightforward to see that AT reduces to the matrix.000000310701TBSo,by Theorem 6,AT and BT have the same
