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 now 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 R$6$ of all possible bar lengths. Does it form a subspace?