Problem 1 (Changing the generating set of a span) This problem has two parts: (a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then v is a linear combination of W1, ..., wn. (b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements. Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn also span V. Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can pick m of them that will still span V?

Illustrate the theorem Matrix Exponential Solutions?

Linear algebra: please show all steps correctly and don't send me previous posted answer

Hello. Please answer the attached Linear Algebra question correctly and completely. Please show all of your work and handwrite your solution.  *If you answer the question correctly and show all of your work, I will give you a thumbs up. Thanks