(4.12.1)--13.InnerProductandEuclideanSpace.ppt
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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
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- 4.12 13. InnerProductandEuclideanSpace