Chapter 4.4, Problem 25E

Exercises 23-26 concern a vector space V, a basis B = {b₁,..., b_n}, and the coordinate mapping x ↦ [ x ]_B.

25. Show that a subset {u₁,...,u_p} in V is linearly independent if and only if the set of coordinate vectors {[ u₁ ]_B,..., [ u_p ]_B} is linearly independent in ℝⁿ. [Hint: Since the coordinate mapping is one-to-one, the following equations have the same solutions, c₁,..., c_p.]

c₁u₁ +    + c_pu_p = 0 The zero vector in V

[c₁u₁ +    + c_pu_p]_B = [0]_B The zero vector in ℝⁿ