Exercise $1.$ Least Squares and QR Factorization

pts$)$

Bob is a city planner working on building a highway that connects $4$ cities. Since

there are no empty spaces between buildings in each city, the highway should

not go through the cities. Instead, it should pass near the cities as much as

possible. The coordinates of the cities are given as follows: $(-2,3),(0,1),(3,0),$

$(5,2).$

You are expected to implement the codes of the following questions in

Python. You should show each question's calculations using the numpy library.

The matrices, equation calculations, and all of the important numerical values

should be clearly demonstrated in the code and printed. Remember to plot all

four roads you have designed separately. Those of you who do not comply with

these requirements will not be able to get full points. You should absolutely

show each step you have performed in your code.

a$.$ Use the least squares method to find a straight road that can be built

according to the requirement given above. $(20pts)$

b$.$ Use the least squares method to find a road with a single curve that can

be built according to the requirement given above. $(25pts)$

c$.$ Calculate the root mean squared error for both roads you have designed

in the previous steps. $(20pts)$

d$.$ Use the classical Gram$-$Schmidt method to find the QR factorization so

that we can improve the design of these roads in such a way that the roads

become even closer to the cities. For the straight road, use reduced QR

factorization. For the curved road, use the full QR factorization. $(35$ pts$)$