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Theorem 4.2LetAbe anm×nmatrixA=a11a12· · ·a1na21a22· · ·a2n............am1am2· · ·amn.and denote the columns ofAby the column vectorsc1,c2, . . . ,cn, so thatci=a1ia2i...ami,1≤i≤n.Then ifx= (x1, x2, . . . , xn)Tis any vector inRn,Ax=x1c1+x2c2+· · ·xncn.This theorem states that the matrix productAx, which is a vector inRm, can beexpressed as a linear combination of the column vectors ofA.Activity 4.3Prove this theorem; derive expressions for both the left-hand side andthe right-hand side of the equality as a singlem×1 vector and compare thecomponents to prove the equality.Readsection 1.8 of the text A-H, working through the activities there. You willfind the solution of the last activity in the text.4.2Developing geometric insight –R2andR3Vectors have a broader use beyond that of being special types of matrices. It is likelythat you have some previous knowledge of vectors; for example, in describing thedisplacement of an object from one point to another inR2or inR3. Before we continueour study of linear algebra it is important to consolidate this background, for itprovides valuable geometric insight into the definitions and uses of vectors in higherdimensions. Parts of this section may be review for you.53
44. Vectors4.2.1Vectors inR2The setRcan be represented as points along a horizontal line, called areal-number line.In order to represent pairs of real numbers, (a1, a2), we use aCartesian plane, a planewith both a horizontal axis and a vertical axis, each axis being a copy of thereal-number line, and we markA= (a1, a2) as apointin this plane. We associate thispoint with the vectora= (a1, a2)T, as representing adisplacementfrom the origin (thepoint (0,0)) to the pointA. In this context,ais theposition vectorof the pointA.This displacement is illustrated by an arrow, or directed line segment, with initial pointat the origin and terminal point atA.-x6y(0,0)>r(a1, a2)a2a1aposition vector,aEven if a displacement does not begin at the origin, two displacements of the samelength and the same direction are considered to be equal. So, for example, the twoarrows below represent the same vector,v= (1,2)T.-x6y(0,0)displacement vectors,vIf an object is displaced from a point, sayO, the origin, to a pointPby thedisplacementp, and then displaced fromPtoQ, by the displacementv, then the totaldisplacement is given by the vector fromOtoQ, which is the position vectorq. So wewould expect vectors to satisfyq=p+v, both geometrically (in the sense of adisplacement) and algebraically (by the definition of vector addition). This is certainlytrue in general, as illustrated below.54