CHANGE 2/17 



transformation U, which will give a transform U{T{n)). Thus, 

 if they are 



_ , a b c d , , a b c d 



T: i and U: i 



b d a b d c d b 



then T{b) is d, and V{T{b)) is U{d), which is b. Tand U appHed in 

 that order, thus define a new transformation, V, which is easily 

 found to be 



^ c b d c 



V is said to be the product or composition of T and U. It gives 

 simply the result of T and U being applied in succession, in that 

 order, one step each. 



If U is apphed first, then U{b) is, in the example above, c, and 

 T{c) is a; so T{U(b)) is a, not the same as U(T(b)). The product, 

 when U and T are applied in the other order is 



'^ b a b d 



For convenience, V can be written as UT, and IV as TU. It must 

 always be remembered that a change of the order in the product may 

 change the transformation. 



(It will be noticed that Kmay be impossible, i.e. not exist, if some 

 of r's transforms are not operands for U.) 



Ex. 1 : Write out in full the transformation U^T. 

 Ex. 2: Write out in full: UTU. 



*Ex. 3 : Represent T and U by matrices and then multiply these two matrices 

 in the usual way (rows into columns), letting the product and sum of +'s 

 be + ; call the resulting matrix Mi. Represent K by a matrix; call it M2. 

 Compare Mi and M2. 



2/17. Kinematic graph. So far we have studied each transforma- 

 tion chiefly by observing its effect, in a single action, on all its 

 possible operands (e.g. S.2/3). Another method (applicable only 

 when the transformation is closed) is to study its effect on a single 

 operand over many, repeated, applications. The method corres- 

 ponds, in the study of a dynamic system, to setting it at some initial 

 state and then allowing it to go on, without further interference, 

 through such a series of changes as its inner nature determines. 

 Thus, in an automatic telephone system we might observe all the 

 changes that follow the dialHng of a number, or in an ants' colony 



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