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

\[ \begin{bmatrix} f_{1} & f_{2} & \ldots & f_{n} \\ f'_{1} & f'_{2} & \ldots & f'_{n} \\ f"_{1} & f"_{2} & \ldots & f"_{n} \\ f_{1}^{n-1} & f_{2}^(n-1) & \ldots & f_{n}^{(n-1)} & \ldots & f_{n}^{(n-1)} \end{bmatrix} \begin{bmatrix} c_{1} \\ c_{2} \\ c_{3} \\ \vdots \\ c_{n} \end{bmatrix} = 0 \]

Now we know that the matrix equation \( AX = \underline{O} \) has a trivial solution if \( |A| \neq 0 \). Hence,

\[ Therefore, c_{1} = c_{2} = \ldots = c_{n} = 0 \]

We have,

\[ \begin{bmatrix} f_{1} & f_{2} & \ldots & f_{n} \\ f'_{1} & f'_{2} & \ldots & f'_{n} \\ f''_{1} & f''_{2} & \ldots & f''_{n} \\ f_{1}^(n-1) & f_{2}^(n-1) & \ldots & \ldots & f_{n}^{(n-1)} & \ldots & f_{n}^{(n-1)} \end{bmatrix} \neq 0 \]

That is,

\[ W(f_{1}, f_{2}, \ldots, f_{n}) \neq 0 \] \n Where W is Wronskian, \n then \( c_{1} = c_{2} = \ldots = c_{n} = 0 \) Therefore, the functions \( f_{1}, f_{2}, \ldots, f_{n} \) are linearly independent over I .

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