I want an algo in python for this and proof of part a. This question if of advanced maths and books like heath can be used for reference

Consider a linearly independent set of n real vectors X1, ..., Xn e R. Suppose another set of vectors Y, ..., Yn R is congruent to it, in the sense that all lengths and distances are equal: ||xi||2 = ||yi||2 for all i, and ||Xi xj|l2 = ||yi yj|l2 for all i + j. Define the matrices X = (X1,...,xn) and Y = (y1,..., yn). (a) Prove that the reduced QR factorizations of X and Y have the same . Hint: What can you say about the relationship between XIX and YTY? (b) Give an algorithm to find an orthogonal matrix Q such that Qxi = y; for all i. Consider a linearly independent set of n real vectors X1, ..., Xn e R. Suppose another set of vectors Y, ..., Yn R is congruent to it, in the sense that all lengths and distances are equal: ||xi||2 = ||yi||2 for all i, and ||Xi xj|l2 = ||yi yj|l2 for all i + j. Define the matrices X = (X1,...,xn) and Y = (y1,..., yn). (a) Prove that the reduced QR factorizations of X and Y have the same . Hint: What can you say about the relationship between XIX and YTY? (b) Give an algorithm to find an orthogonal matrix Q such that Qxi = y; for all