Prove that if the Wronskian of the functions are Linearly Independent over I
Th. Prove that if the Wronskian of the functions \(f_{1}\, f_{2}\, \ldots f_{n}\) over an Interval I is non-zero, then the functions are linearly independent over I. Proof Proof: Consider the relation \[ c_{1}f_{1} + c_{2}f_{2} + \ldots + c_{n}f_{n} = 0 ...(1)\] where \( z_{1}, c_{2}, \ldots, c_{n} \) are constants. Differentiating (1) successively \( n-1 \) times with respect to \( x \), we get, \[ c_{1}f'_{1} +c_{2}f'_{2} +c_{3}f'_{3} + \ldots + c_{n} {f_{n}}^{\prime} = 0 ...(2)\] \[ c_{1}f''_{1} + c_{2}f''_{2} + \ldots + c_{n}f''_n = 0 ...(3)\] \[ c_{1}{f_{1}}^{(n-1)} + c_{2}{f_{2}}^{(n-1)} + \ldots + c_{n}{f_{n}}^{(n-1)} = 0 ...(n)\] Here The Vidoe in Youtube These \( n \) equations can be written as \[ \begin{bmatrix} f_{1} & f_{2} & \ldots & f_{n} \\ f'_{1} & f'_{2} & \ldots & f'_{n} \\ f"_{1} & f"_...