Question

Which one of the following statement is not correct?

• A. The functions \( x^2-1, 3x^2 \) and \( 2-5x^2 \) are linear dependent.
• B. The functions \( x, x^2 \) and \( x^3 \) are linearly independent.
• C. The functions 1, sinx and cosx are linearly dependent.
• D. The functions x and \( \frac{1}{x} \) are linearly independent.

Hint:

To quickly test linear independence, consider if one function can be written as a linear combination of the others. In (A), you can see that \(2(x^2-1) + 1(3x^2) = 5x^2-2\), which is \(-1 \times (2-5x^2)\). So \(2f_1 + f_2 + f_3 = 0\). For (C), it's impossible to write, for example, \(\sin x\) as \(c_1(1) + c_2(\cos x)\) for all x.

Answer 1

The Correct Option is C

Step 1: Definition of Linear Dependence:
A collection of functions \( f_1, f_2, \ldots, f_n \) is linearly dependent if there exist non-zero constants \( c_1, c_2, \ldots, c_n \) such that their linear combination \( c_1 f_1(x) + c_2 f_2(x) + \ldots + c_n f_n(x) = 0 \) for all values of x in their common domain. If no such non-zero constants exist, the functions are linearly independent. For differentiable functions, linear independence can also be determined by a non-zero Wronskian at any point in their domain.
Step 2: Analysis of Examples:

(A) \( f_1=x^2-1, f_2=3x^2, f_3=2-5x^2 \): We seek constants \(c_1, c_2, c_3\), not all zero, for which \( c_1(x^2-1) + c_2(3x^2) + c_3(2-5x^2) = 0 \). Rearranging by powers of x yields \( (c_1 + 3c_2 - 5c_3)x^2 + (-c_1 + 2c_3) = 0 \). For this equality to hold for all x, the coefficients must be zero: \( c_1 + 3c_2 - 5c_3 = 0 \) \( -c_1 + 2c_3 = 0 \implies c_1 = 2c_3 \). Substituting \(c_1\) into the first equation: \( (2c_3) + 3c_2 - 5c_3 = 0 \implies 3c_2 - 3c_3 = 0 \implies c_2 = c_3 \). Choosing \( c_3 = 1 \) yields \( c_2 = 1 \) and \( c_1 = 2 \). Since a non-trivial solution (2, 1, 1) exists, the functions are linearly dependent. The statement is correct.
(B) \( x, x^2, x^3 \): For the equation \( c_1x + c_2x^2 + c_3x^3 = 0 \) to hold for all x, all coefficients must be zero (\(c_1=c_2=c_3=0\)), as a non-zero polynomial has a finite number of roots. Therefore, the functions are linearly independent. The statement is correct.
(C) 1, sinx, cosx: Consider the equation \( c_1(1) + c_2\sin x + c_3\cos x = 0 \). For this to be true for all x, we must have \(c_1=c_2=c_3=0\). Evaluating at specific points: at \(x=0\), \(c_1+c_3=0\); at \(x=\pi/2\), \(c_1+c_2=0\); at \(x=\pi\), \(c_1-c_3=0\). From \(c_1+c_3=0\) and \(c_1-c_3=0\), we deduce \(c_1=0\) and \(c_3=0\), which in turn implies \(c_2=0\). The only solution is the trivial one. Thus, the functions are linearly independent. The statement claims they are linearly dependent, making it not correct.
(D) x and \( \frac{1}{x} \): The equation \( c_1x + c_2(1/x) = 0 \) can be rewritten as \( c_1x^2 + c_2 = 0 \) by multiplying by x. For this to hold for all x, we must have \(c_1=0\) and \(c_2=0\). Hence, the functions are linearly independent. The statement is correct.
Step 3: Conclusion:
Statement (C) is incorrect because the functions 1, sinx, and cosx are linearly independent.