By Wai-Kai Chen, Elektroingenieur

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Now, consider a mapping b from U × V into W. For brevity of notation, this can be written b: U × V → W. 3) for all f1 and f2 in F, for all u, u1, and u2 in U, and for all v, v1, and v2 in V. The basic idea of the bilinear operator is apparent from this definition. It is an operator with two arguments, having the property that if either of the two arguments is fixed, the operator becomes linear in the remaining argument. A moment’s reflection will show that the intuitive operation of multiplication is of this type.

It is easy to check that these choices produce the correct first three coordinates; the last coordinate is 4h3h2/9h1, which by virtue of the property 9h1h4 = 4h2h3 is equal to h4 as desired. Thus, a vector in W is in the image of b if and only if the relation 9h1h4 = 4h2h3 is satisfied. book Page 26 Wednesday, September 18, 2002 10:57 PM 26 The Circuits and Filters Handbook, Second Edition shown that the vectors in this class are not closed under addition. For this purpose, simply select a pair of vectors represented by (1, 1, 9, 4) and (4, 9, 1, 1).

Other than being a description in terms of inputs and outputs, as above, linear systems may also be described in terms of sets of other types of equations, such as differential equations or difference equations. When that is the situation, the familiar notion of initial condition becomes an instance of the idea of state, and one must examine the intertwining of states and inputs to give outputs. 50), when each input yields a unique output, is said to define a system function. If the differential (difference) equations are linear and have constant coefficients, the possibility exists of describing the system in terms of transforms, for example, in the s-domain or z-domain.