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07_hwc1SolnsODDA

# 07_hwc1SolnsODDA - 2.3 Consider the following...

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2.3 Consider the following transformations from IR 2 to IR 2 . 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 = x 2 - y 2 x 2 + 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: First, for any number a , and any vector x = x y , f a x y = f ax ay = ax + ay ay - ax = a x + y y - x = af x y . Hence f is homogenous. However, g a x y = g ax ay = | a || x | ay - ax If a > 0, so | a | = a , then this equals a | x | y - x = ag x y but otherwise not. Since the homogenity requires equality for
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