Explanation to Problem B1:

The dimensionality as we observe from the residual variance curve is 2. The dimensionality is easily verified as in the images the robot's upper arm moves in the horizontal direction , hence the upper arm has its independent angle , while the lower arm has it own angular movement along the joint. Hence, the theta1 and theta2 of the two arms,respectively form the two degrees of freedom of  the images. However, in some cases the degrees of freedom may come to 3 if we increase the number of neighbourhood points in the graph. This dimensionality can be thought of the up down movement through the shoulder joint.