Task 2: Properties of Orthogonal Matrices

Given that an orthogonal matrix Q satisfies QᵀQ = I, verify this property for the matrix Q obtained in Task 1.

Task 3: Application of QR Decomposition

Explain how QR decomposition can be used in solving linear systems, specifically referring to the computational advantages it provides over other methods.

Task 1: Perform QR decomposition on matrix A using the Gram-Schmidt process to find matrices Q and R.

A = [ 1 1

        1 0

        1 -1] (3x2)

Task 4: Practical Implementation of QR Decomposition

Write a Python function that takes any matrix and returns the QR decomposition of that matrix using NumPy's linear algebra library. Include comments in your code to explain each step.