2.3Consider the following transformations fromIR2toIR2. Which ones are linear? Explain your answers,and for those that are linear, write down the corresponding matrix.fxy=y+xy-xgxy=|x|y-xhxy=x2-y2x2+y2SOLUTIONThe only linear transformation isf. To see this, we check homogeneity andadditivity. In fact, it turns out that onlyfis homogeneous:First, for any numbera, and any vectorx=xy,faxy=faxay=ax+ayay-ax=ax+yy-x=afxy.Hencefis homogenous.However,gaxy=gaxay=|a||x|ay-axIfa >0, so|a|=a, then this equalsa|x|y-x=agxybut otherwise not. Since the homogenity requires equality for
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