Linearity 1. Show that the function L = x - z is a linear function, using the definition of 'linear'.1 2. Show that the function L = xy is not linear, using the definition of 'linear'. 3. Determine whether or not the function L prove it. = x + y + 1 is a linear function. If it is, prove it; if not, 4. Use this applet to visualize the transformation A mation does.2 ( 2 + 3) . Describe in words what this transfor- 5. I mentioned in class that rotation counter-clockwise by 0 radians is a linear transformation Re : R2 - 2. In this problem, you will find a formula for this linear transformation. a) Take the point in the plane (one unit right on the x-axis). Explain why rotating this point 0 radians counter-clockwise gives the point (cos e sin 0 b). Take the point (9) in the plane (one unit up on the y-axis), Explain why rotating this point & radians counter-clockwise gives the point (cos ( 0 + TT / 2) ) sin (0 + 7 /2) ) Some high-school trig shows that this is equal to -sin(0)) cos(0) c) In the previous two parts, you showed that Re cos e sin 0 Re cos e Use the assumption that Re is linear to show that Re x cos 0 - y sin 0 z sin ( + y cos e ) 1 A function is linear if it satisfies L(v + w) = L(v) + L(w) for all pairs of vectors, and furthermore L(cv) = cL(v) for all scalars c. "There is a description of how to use this applet in this week's notes. If it helps, there is a way to have Desmos show you what happens gradually: click next to amin in the bottom line and change 'step' to 0.1 instead of 1