235_course_notes_chapter3 - Chapter 3 Inner Products In...

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Chapter 3Inner ProductsIn Linear Algebra 1, we briefly looked at the dot product function for vectors inRn.We saw that the dot product function has an important relationship to the lengthof a vector inRnand to the angle between two vectors. Moreover, the dot producthas some important applications; in particular, it gave us an easy way of finding theprojection of a vector onto a plane. Our goal in this chapter is to extend these ideasto general vector spaces.3.1Inner Product SpacesRecall that we defined a vector space so that the operations of addition and scalarmultiplication had the essential properties of addition and scalar multiplication ofvectors inRn. Thus, to generalize the concept of the dot product it makes sense toinclude the essential properties of the dot product in our definition.DEFINITIONInner ProductLetVbe a vector space. Aninner productonVis a function(,):V×VRthat has the following properties: For everyvectorv, vectoru, vectorwVands, tRwe haveI1(vectorv,vectorv) ≥0 and(vectorv,vectorv)= 0 if and only ifvectorv=vector0(Positive Definite)I2(vectorv, vectorw)=(vectorw,vectorv)(Symmetric)I3(svectorv+tvectoru, vectorw)=s(vectorv, vectorw)+t(vectoru, vectorw)(Bilinear)DEFINITIONInner ProductSpaceA vector spaceVwith an inner product(,)onVis called aninner product space.In the same way that a vector space is dependent on the defined operations of additionand scalar multiplication, an inner product space is dependent on the definitions ofaddition, scalar multiplication, and the inner product.So, when defining an innerproduct space, one must specify which inner product is being used.1
2Chapter 3Inner ProductsEXAMPLE 1The dot product (often called the standard inner product) defines an inner productonRn.EXAMPLE 2Which of the following defines an inner product onR3?(a)(vectorx, vectory)=x1y1+ 2x2y2+ 4x3y3.

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