Problem 2. The robotic arm NOOKLOOK is a tool that is designed to look for small metallic objects in a difficult landscape. The arm has the following structure: it is based on a platform, and all the joints are rigid (can not flex or rotate), but the bars can telescopically retract and extend. The bars can even retract all the way to the opposite direction than the one they are facing without changing the direction of the next bar in the arm. The end of the arm has a small magnet on it that allows to pick up metallic objects. The geometry of the bars and the robot movement is presented on the following pictures: The directions of the bars, in succession, are: (0,0,1), (1,0,1), (0,1,1), (1,1,1), (1, 1, −1), and (0, 0, -1). (1) Consider the transformation T that inputs the current lengths of the bars and outputs the position of the end of the arm. What are the domain and the codomain of T? Explain why T is linear. Can you provide its matrix A? (2) Is this transformation injective? Is this transformation surjective? Justify. Interpret your an- swers in terms of the robotic arm manipulations. (3) You are searching for a coin (of negligible size) that is lying on an incline surface given by the condition x + y - z = 2. The robotic arm is very fragile, so you wouldn't want it to bump into the surface, but you also need the end to go along the surface in order to be sure that you find the coin. What restrictions would you impose on the lengths of the bars for this search? For simplicity, assume that all the parts of the arm can pass through the incline surface and only the end is restricted to be on the surface. While exploring this problem you can use MATLAB for row reduction. However your final answer should be done by hand. (4) Consider the set of all admissible (for the previous question) lengths of the bars as a subset in the set R6 of all possible bar lengths. Does it form a subspace? Definition 3. An (n − 1)-dimensional subspace of R is called a hyperplane in Rn. (5) A plane in R³ is an example of a hyperplane. Show that the equation C₁x1 + €2x2 + C3x3 + €4x4 + C5x5+C6x6 = 0 defines a hyperplane in R6. (6) Assume now that the incline surface P₁ is given by x + y − z = 0. A corner is formed by this plane, the plane P2 defined by y − z = = 0, and the floor P3, which is given by z = 0. What subset of R6 describes admissible bar lengths for the arm end to belong to each of these three planes individually? Explain why are these subsets subspaces in R6 and describe them as span of vectors in R6. Does any of them make a hyperplane in R6? If yes, describe the hyperplane using a single linear equation. You can use Geogebra or MATLAB when exploring this problem, but your final answer should not depend on your observations in Geogebra or MATLAB. Definition 4. Given a function f: A → B and a subset DC B, the preimage of D under f is defined as f¹(D) = {a ЄA | ƒ(a) € D}. That is, the preimage of D is the set of all elements in the domain whose images are in D. Note that preimage is defined regardless of whether or not f is invertible. If f is invertible the preimage of a single element bЄ B is the same as the output of b under the inverse map f-1, both denoted by f¯¹(b). In the previous part you found the preimage of P1, P2 and the floor P3 under T. That is, you found T-¹(P1), T-¹(P2) and T-11 -1 (P3). (7) What subset of R6 describes admissible bar lengths for the arm end to belong to the line formed by intersecting the two incline planes P₁ x + y − z = 0 and P2 y − z = 0. Based on your observation, what is the relation between T-¹ (P1 P2) and T¯¹(P₁) NT¯¹(P2)? (8) What is are the possible dimensions of the intersections of two hyperplanes in R6? What are the possible dimensions of the intersection of two hyperplanes in Rn? you (9) Now let's go back to NOOKLOOK and consider the line segment formed by intersecting the two incline planes x + y - z = 0 and y - z = 0, bounded by the floor z = 0. Recall that NOOKLOOK's arm end is very sensible and we would like to avoid the end hitting the floor. How would adjust your admissible bar lengths from the previous parts to give a set of admissible bar lengths so that the arm end explore this intersection line and stops when it hits the corner made with the floor?