1、Linear AlgebraInner Product andEuclidean Space Inner ProductInner Product For any two vectors in Rn,x=(x1,.,xn)T and y=(y1,.,yn)T,the product xTy is called the scalar product in Rn,where xTy=x1y1+x2y2+xnyn.Then we can define the length|x|of vector x and the angle between nonzero vectors x and y in t
2、erms of the scalar product,such as,|x|=(xTx)1/2 and cos=xTy/(|x|y|).Further,we can describe the structure of vector space Rn.Scalar products are useful not only in Rn,but in a wide variety of contexts.To generalize this concept to other vector spaces,we introduce the following definition.Definition
3、Let V be a vector space on the real number field R,an inner product on V is an operation that assigns,to each pair of vectors and in V,a real number ,satisfying the following conditions for any vectors ,in V and kR:(1),=,;(2)+,=,+,(3)k,=k,(4),0,and ,=0 if and only if =0.The real linear space with in
4、ner product is refered to the Euclidean space.Example1.Rn with the standard inner product ,=T=x1y1+.+xnyn is a Euclidean space.If another inner product is defined as,=x1y1+2x1y1.+nxnyn,then Rn is another Euclidean space.2.Ca,b with the following inner product is a Euclidean space.3.Rmn with the foll
5、owing inner product is a Euclidean space.f(x),g(x)Ca,b.By using the concept of inner product,we can define the length and angle of vectors in Euclidean space.Length and Angle of vectorsLength and Angle of vectorsDefinition Suppose V is a Euclidean space,V,the length(or norm)of is defined as|=,1/2(or
6、 denoted by|).Remarks:(1)is called unit vector,if|=1;(2)if 0,then(1/|)is a unit vector;(3)Cauchy-Schwarz inequality|,|.Properties:(1)|0,with equality if and only if =0;(2)|k|=|k|;(3)|+|+|.Definition Suppose V is a Euclidean space,the angle between two nonzero vectors and in V is defined as From the
7、definition,we know ,=|cos=0 if and only if=/2.Definition The vectors and in V are said to be orthogonal if ,=0.Example In the Euclidean space R3 with the standard inner product x,y=x1y1+x2y2+xnyn,find a unit vector such that it is orthogonal to 1=(1,1,1)T,2=(0,1,1)T.Solution First find a vector =(x1,x2,x3)T which is orthogonal to 1 and 2.=Then,find the unit vector The last process is called unitization.
