线性代数教学资料-cha.ppt
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1、3 The Vector Space Rn,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 Subspaces,Core Sections,In mathematics and the physical sciences,the term vector is applied to a wide variety of objects.Perhaps the most familiar applicat
2、ion 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 such as matrices,poly
3、nomials,and continuous real-valued functions.,3.1 Introduction,In 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.,3.2 VECTOR SPACE
4、PROPERTIES OF Rn,The Definition of Subspaces of Rn,A 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 any scalar.,Example 1:Let W be the subset of R3 de
5、fined by,Verify 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 whether a vector in Rn is or is not in W.Step 2.Test the zero vector,of Rn to see whether
6、 it satisfies the algebraic specification required to be in W.(This shows that W is nonempty.),Verifying that W is a subspace of Rn,Step 3.Choose two arbitrary vectors X and Y from W.Thus X and Y are in Rn,and both vectors satisfy the algebraic specification of W.Step 4.Test the sum X+Y to see wheth
7、er 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,Show that W is not a subspace of R3.,Example 2:Let W be the subset of R3 defined by,Verify that W is a subspa
8、ce of R3 and give a geometric interpretation of W.,Example 4:Let W be the subset of R2 defined by,Demonstrate that W is not a subspace of R2.,Example 5:Let W be the subset of R2 defined by,Demonstrate that W is not a subspace of R2.,Exercise P175 18 32,3.3 EXAMPLES OF SUBSPACES,In this section we in
9、troduce several important and particularly useful examples of subspaces of Rn.,The span of a subset,Theorem 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 combin
10、ations of v1,vr is called the subspace spanned by S and will be denoted by Sp(S)or Spv1,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,a
11、nd are not on the same space,then Spu,v,w=au+bv+cw:a,b,c any real numbers,Example 1:Let u and v be the three-dimensional vectors,Determine W=Spu,v and give a geometric interpretation of W.,The null space of a matrix,We now introduce two subspaces that have particular relevance to the linear system o
12、f equations Ax=b,where A is an(mn)matrix.The first of these subspaces is called 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
13、 an(m n)matrix,then N(A)is a subspace of Rn.,Example 2:Describe N(A),where A is the(3 4)matrix,Solution: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,Solving the corresp
14、onding reduced system yields,x1=-2x3-3x4 x2=-x3+2x4,Where x3 and x4 are arbitrary;that is,Example 5:Let S=v1,v2,v3,v4 be a subset of R3,where,Show that there exists a set T=w1,w2 consisting of two vectors in R3 such that Sp(S)=Sp(T).,Solution:let,Set row operation to A and reduce A to the following
15、matrix:,So,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,The solution on P184,And 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,So,by Theorem 6,AT and BT h
16、ave the same row space.Thus A and B have the same column space where,In particular,Sp(S)=Sp(T),where T=w1,w2,Two of the most fundamental concepts of geometry are those of dimension and the use of coordinates to locate a point in space.In this section and the next,we extend these notions to an arbitr
17、ary subspace of Rn by introducing the idea of a basis for a subspace.,3.4 BASES FOR SUBSPACES,An example from R2 will serve to illustrate the transition from geometry to algebra.We have already seen that each vector v in R2,can be interpreted geometrically as the point with coordinates a and b.Recal
18、l that in R2 the vectors e1 and e2 are defined by,Clearly the vector v in(1)can be expressed uniquely as a linear combination of e1 and e2:v=ae1+be2(2),As we will see later,the set e1,e2 is an example of a basis for R2(indeed,it is called the natural basis for R2).In Eq.(2),the vector v is determine
19、d by the coefficients a and b(see Fig.3.12).Thus the geometric concept of characterizing a point by its coordinates can be interpreted algebraically as determining a vector by its coefficients when the vector is expressed as a linear combination of“basis”vectors.,Spanning sets Let W be a subspace of
20、 Rn,and let S be a subset of W.The discussion above suggests that the first requirement for S to be a basis for W is that each vector in W be expressible as a linear combination of the vectors in S.This leads to the following definition.,Definition 3:Let W be a subspace of Rn and let S=w1,wm be a su
21、bset of W.we say that S is a spanning set for W,or simply that S spans W,if every vector w in W can be expressed as a linear combination of vectors in S;w=a1w1+amwm.,A restatement of Definition 3 in the notation of the previous section is that S is a spanning set of W provided that Sp(S)=W.It is evi
22、dent that the set S=e1,e2,e3,consisting of the unit vectors in R3,is a spanning set for R3.Specifically,if v is in R3,Then v=ae1+be2+ce3.The next two examples consider other subset of R3.,Example 1:In R3,let S=u1,u2,u3,where,Determine whether S is a spanning set for R3.,Solution:The augmented matrix
23、,this matrix is row equivalent to,Example 2:Let S=v1,v2,v3 be the subset of R3 defined by,Does S span R3?,Solution:,and the matrix A|v is row equivalent to,So,is in R3 but is not in Sp(S);that is,w cannot be expressed as a linear combination of v1,v2,and v3.,The next example illustrates a procedure
24、for constructing a spanning set for the null space,N(A),of a matrix A.Example 3:Let A be the(34)matrix,Exhibit a spanning set for N(A),the null space of A.,Solution:The first step toward obtaining a spanning set for N(A)is to obtain an algebraic specification for N(A)by solving the homogeneous syste
25、m Ax=.,Let u1 and u2 be the vectors,Therefore,N(A)=Spu1,u2,Minimal spanning sets If W is a subspace of Rn,W,then spanning sets for W abound.For example a vector v in a spanning set can always be replaced by av,where a is any nonzero scalar.It is easy to demonstrate,however,that not all spanning sets
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