Unformatted text preview: If g were reﬂection about the y axis, we would have Ag◦f = Ag Af = 1 1 + s2 −1 0 0 1 1 − s2 2s 1 2s = s −1 1 + s2
2 s2 − 1 −2s 2s s2 − 1 . 2.17 Consider the function from IR2 to IR2 deﬁned by f f x y = x3 /(x2 + y 2 ) . − y 3 ( x2 + y 2 ) 0 0 = 0 , and in case 0 x y = 0 , 0 (a) Show that for all x in IR2 and all numbers a, f (ax) = af (x). (b) Show that f is not linear. SOLUTION (a) If x = y = 0, or if a = 0, af (x) = f (ax) = 0. So suppose that a = 0, and that at least one of x and y is non zero. Then (ax)3 /((ax)2 + (ay )2 ) is well deﬁned, and cancelling two powers of a, equals ax3 /(x2 + y 2 ). Likewise with y 3 (x2 + y 2 ), and thus f (ax) = af (x). That is, f is homogeneous. (b) We compute f however f Since f 1 0 +f 0 1 =f 1 1 , 1 1 = 1 1 2 −1 . 1 0 = 1 0 and f 0 1 = 0 −1 , f is not additive, and hence not linear.
2.19 Determine whether there is a linear transformation from IR2 to IR2 such that f 1 0 = 2 3 f 0 1 = 3 2 and f 1 1 = 5 5 If so, ﬁnd the matrix of such a transformation. If not, explain why not. SOLUTION If there is such a linear transformation f , it must be given by the matrix 23 Af = , where we have used the ﬁrst two pieces of information given about f , and 32 Theorem 2.
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 Fall '08
 Gladue
 Calculus, Derivative, Vector Space, Linear map, Linear function, IR2

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