The question is attached in the image. How can a function defined in the following manner be 'linearly' independent? The function is given by "fk (x) = x^k". For any x value we can define a transformation in this manner but why will the set be 'linearly' independent?Please provide a solution for the question given in the image. Thank you. 11. The set {f1, f2, ..., fn} where each fr is a real-valued function defined on R, is said tobe linearly independent if c1, C2, . .. , Cn E R and EC fr (x) = 0 for every r ER impliesC1 = C2 = ·….= Cn = 0Suppose fr(x) = x* for all r E R and k = 1,..., n. Then,A. the set {f1, ..., fn} is linearly independent.B. each pair of these functions is linearly independent, but larger n-tuples are not.C. only the subset of odd-numbered functions and the subset of even-numberedfunctions are linearly independent.D. every "proper subset of this set of functions is linearly independent, but thewhole set is not.

Answered: Image /qna-images/answer/6cde3fbc-7a7e-4e47-b73f-56e87d17dc74.jpg

11. The set {f, f2, . .., fn} where each fk is a real-valued function defined on R, is said to be linearly independent if c1, C2, . .. , Cn E R and E C fk (x) = 0 for every r ER implies C1 = c2 = ·..= Cn = 0 Suppose fr(x) = x* for all x ER and k = 1, ..., n. Then, A. the set {f1,..., fn} is linearly independent. B. each pair of these functions is linearly independent, but larger n-tuples are not. C. only the subset of odd-numbered functions and the subset of even-numbered functions are linearly independent. D. every "proper subset of this set of functions is linearly independent, but the whol