Linear Algebra

Topic: Dimensional Analysis 155(b) Show that {Π 1 = h/d, Π 2 = dg/v 2 } is a complete set of dimensionless products.(c) Show that if h/d is fixed then the propagation speed is proportional to thesquare root of d.5 Prove that the dimensionless products form a vector space under the ⃗+ operationof multiplying two such products and the ⃗· operation of raising such the productto the power of the scalar. (The vector arrows are a precaution against confusion.)That is, prove that, for any particular homogeneous system, this set of products ofpowers of m 1 , . . . , m kis a vector space under:{m p 11 . . . mp kk∣ p1 , . . . , p k satisfy the system}m p 11 . . . mp k ⃗ k+m q 11 . . . mq kk = mp 1+q 11. . . m p k+q kkandr⃗·(m p 11 . . . mp kk ) = mrp 11. . . m rp kk(assume that all variables represent real numbers).6 The advice about apples and oranges is not right. Consider the familiar equationsfor a circle C = 2πr and A = πr 2 .(a) Check that C and A have different dimensional formulas.(b) Produce an equation that is not dimensionally homogeneous (i.e., it addsapples and oranges) but is nonetheless true of any circle.(c) The prior item asks for an equation that is complete but not dimensionallyhomogeneous. Produce an equation that is dimensionally homogeneous but notcomplete.(Just because the old saying isn’t strictly right, doesn’t keep it from being auseful strategy. Dimensional homogeneity is often used to check the plausibilityof equations used in models. For an argument that any complete equation caneasily be made dimensionally homogeneous, see [Bridgman], Chapter I, especiallypage 15.)