(Solved): 4) Consider the vector space M22 of all 2 2 matrices with normal matrix addition and scalar mul ...

4) Consider the vector space M22 of all 2 × 2 matrices with normal matrix addition and scalar multiplication as the operations. Answer the following questions: T T 1 4.1) Let W be the set of all matrices A € M?2 such that (4 [¹3])² = 4² [¹ A Prove or disprove that W is a subspace of M22. 4.2) Let A? [30].42-3-4-[68] ? A? - A3 and A? be matrices in [a] M22. Answer the following questions: 4.2.1) For which values of ? will S = {A1, A2, A3, A4} form a linearly independent set? Show all calculations. (2) 4.2.2) Let B = [aa] be a matrix in M22. Set up a linear system of equations that need to be satisfied such that B is spanned by the vectors in S. (1) 4.2.3) Use the system in question 4.2.2 to find out if there are values of ? for which {A1, A2, A3, A4} will be a basis for M22. If there are such values, find the set S = them all. (2) - 2