Question

Which of the following options represent(s) linearly independent pair(s) of functions of a real variable $x$?

• A. $e^{ix}$ and $e^{-ix}$
• B. $x$ and $e^x$
• C. $2^x$ and $2^{-3+x}$
• D. $e^{ix}$ and $\sin x$

Hint:

Two functions are linearly independent if no constant multiple of one function can express the other.

Answer 1

The Correct Option is A, B, D

- (A) $e^{ix$ and $e^{-ix}$:} These are linearly independent, as they are complex exponentials with different exponents.
- (B) $x$ and $e^x$: These functions are linearly independent because one is a polynomial and the other is an exponential.
- (C) $2^x$ and $2^{-3+x$:} These are linearly dependent because $2^{-3+x}$ can be written as a constant multiple of $2^x$.
- (D) $e^{ix$ and $\sin x$:} These are linearly independent. $\sin x$ can be written as $\frac{e^{ix} - e^{-ix}}{2i}$, so they are linearly independent.
Thus, the correct answer is (A), (B), (D).