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Unformatted text preview: 2.3 Consider the following transformations from IR2 to IR2 . Which ones are linear? Explain your answers, and for those that are linear, write down the corresponding matrix. f x y = y+x y−x g x y = |x| y−x h x y = x2 − y 2 x2 + y 2 SOLUTION The only linear transformation is f . To see this, we check homogeneity and additivity. In fact, it turns out that only f is homogeneous: x First, for any number a, and any vector x = , y f a x y =f ax ay = ax + ay x+y =a = af ay − ax y−x x y . Hence f is homogenous. However, ga x y =g ax ay = |a||x| ay − ax If a > 0, so |a| = a, then this equals a |x| = ag y−x x y but otherwise not. Since the homogenity requires equality for every a, g is not homogeneous, and therefore not linear. Also, ha x y =h ax ay = a2 x2 − a2 y 2 a2 x2 + a2 y 2 = a2 x2 − y 2 x2 + y 2 = a2 h x y . Taking any value of a other than a = 1, we see that h is not homogeneous. (It looks almost homogeneous, and functions that behave like h are sometimes called “homogeneous
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