By Robin Chapman
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Extra info for A Guide to Arithmetic [Lecture notes]
Let X = ( ~ ) and P(X) = (;:). and calculate x' and y' in terms of x and y. 1 25 Transformations of the Plane (a) U = (b) U (~), =(~), (c) U = ( _I I)' (d)U=n)· In each case draw a diagram and indicate several vectors and their images. Exercise 2. Consider the line 5x - 2y = 0 and let P denote projection on this line. p( ~), express a' and b' in terms of a and b. If (~) is a given vector and (~:) = Exercise 3. For each of the vectors U in Exercise I, let S(X) = (~:) denote the reflection of (; ) in the line along U.
Conversely, suppose ad - be =1= 0. Let X = (;) be a vector with A (X) = O. Then (0)O = A (X) = (a b) ( x ) = (ax + bY). So ed Y ex + dy + by = 0, ex + dy - 0. ax Multiplying the first equation by d and the second by b and subtracting, we get ( ad - be)x = 0, and hence x =0. Similarly, we get y = 0. Hence X =(;) =(~). 4 Inverses and Systems of Equations x = 0 is the only vector with A (X) = 0, so (7) holds. The proposition is proved. We saw earlier that if A has an inverse, then (7) holds and so ad - bc =F 0.
We need the linear transformation which is the analogue of the function f(x) = x. That function sends every number into itself. The identity transformation, denoted I, sends every vector into itself: I (X) = X, for every vector X. 2 31 Linear Transformations and Matrices Since I sends X = (;) into I(X) = (;), the system x'=x, y'=y describes I. Thus the matrix of I is m(l)=(6 (vi) ~). Finally, we need the linear transformation zero, denoted 0, which sends every vector into the zero vector: O(X) = 0, for all X.