Linear Algebra

146 Chapter Two. Vector SpacesTopic: Dimensional Analysis“You can’t add apples and oranges,” the old saying goes. It reflects our experiencethat in applications the quantities have units and keeping track of thoseunits is worthwhile. Everyone has done calculations such as this one that usethe units as a check.60 sec min hr daysec· 60 · 24 · 365 = 31 536 000min hr day year yearHowever, the idea of including the units can be taken beyond bookkeeping. Itcan be used to draw conclusions about what relationships are possible amongthe physical quantities.To start, consider the physics equation: distance = 16 · (time) 2 . If thedistance is in feet and the time is in seconds then this is a true statement aboutfalling bodies. However it is not correct in other unit systems; for instance, itis not correct in the meter-second system. We can fix that by making the 16 adimensional constant.dist = 16ftsec 2 · (time)2For instance, the above equation holds in the yard-second system.distance in yards = 16(1/3) ydsec 2 · (time in sec) 2 = 16 3yd · (time in sec)2sec2 So our first point is that by “including the units” we mean that we are restrictingour attention to equations that use dimensional constants.By using dimensional constants, we can be vague about units and say onlythat all quantities are measured in combinations of some units of length L,mass M, and time T . We shall refer to these three as dimensions (these are theonly three dimensions that we shall need in this Topic). For instance, velocitycould be measured in feet/second or fathoms/hour, but in all events it involvessome unit of length divided by some unit of time so the dimensional formulaof velocity is L/T . Similarly, the dimensional formula of density is M/L 3 . Weshall prefer using negative exponents over the fraction bars and we shall includethe dimensions with a zero exponent, that is, we shall write the dimensionalformula of velocity as L 1 M 0 T −1 and that of density as L −3 M 1 T 0 .In this context, “You can’t add apples to oranges” becomes the advice tocheck that all of an equation’s terms have the same dimensional formula. An exampleis this version of the falling body equation: d − gt 2 = 0. The dimensionalformula of the d term is L 1 M 0 T 0 . For the other term, the dimensional formulaof g is L 1 M 0 T −2 (g is the dimensional constant given above as 16 ft/sec 2 )and the dimensional formula of t is L 0 M 0 T 1 , so that of the entire gt 2 term isL 1 M 0 T −2 (L 0 M 0 T 1 ) 2 = L 1 M 0 T 0 . Thus the two terms have the same dimensionalformula. An equation with this property is dimensionally homogeneous.Quantities with dimensional formula L 0 M 0 T 0 are dimensionless. For example,we measure an angle by taking the ratio of the subtended arc to theradius