Let A be a linear transformation. If z is the center of the straight interval [x, y], show that Az is the center of the interval [Ax, Ay]. Hint: What does it mean that z is the center of the interval [x, y]?
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Following the hint, it means that z = (x+y)/2. So
Az = A[(x+y)/2]
= A[(1/2) x + (1/2) y]
= A[(1/2) x] + A[(1/2) y]
[explanation: since A is a linear transformation, A(u+v) = Au + Av for all u and v. We apply this here with u = (1/2) x and v = (1/2) y.]
= (1/2) Ax + (1/2) Ay
[explanation: since A is a linear transformation, A(ku) = k Au for all scalars k and all u. We apply this here with k = 1/2 and u = x in the first term, and k = 1/2 and u = y in the second.]
and of course (1/2) Ax + (1/2) Ay is the midpoint of the segment with endpoints Ax and Ay.
Az = A[(x+y)/2]
= A[(1/2) x + (1/2) y]
= A[(1/2) x] + A[(1/2) y]
[explanation: since A is a linear transformation, A(u+v) = Au + Av for all u and v. We apply this here with u = (1/2) x and v = (1/2) y.]
= (1/2) Ax + (1/2) Ay
[explanation: since A is a linear transformation, A(ku) = k Au for all scalars k and all u. We apply this here with k = 1/2 and u = x in the first term, and k = 1/2 and u = y in the second.]
and of course (1/2) Ax + (1/2) Ay is the midpoint of the segment with endpoints Ax and Ay.