Linearly Dependent Functions — Definition, Formula & Examples

Linearly dependent functions are functions where at least one can be written as a combination of the others using constant multipliers. In other words, there exist constants (not all zero) such that a weighted sum of the functions equals the zero function everywhere.

A set of functions $\{f_1, f_2, \ldots, f_n\}$ defined on an interval $I$ is linearly dependent on $I$ if there exist constants $c_1, c_2, \ldots, c_n$ , not all zero, such that $c_1 f_1(x) + c_2 f_2(x) + \cdots + c_n f_n(x) = 0$ for all $x \in I$ . If the only solution is $c_1 = c_2 = \cdots = c_n = 0$ , the functions are linearly independent.

Key Formula

W(f_1, f_2) = \begin{vmatrix} f_1 & f_2 \\ f_1' & f_2' \end{vmatrix} = f_1 f_2' - f_2 f_1'

Where:

$f_1, f_2$ = The two functions being tested
$f_1', f_2'$ = First derivatives of the functions
$W$ = Wronskian determinant; if zero everywhere, functions may be dependent

How It Works

To check whether functions are linearly dependent, you can compute the Wronskian determinant. Form a matrix whose rows contain each function and its successive derivatives, then take the determinant. If the Wronskian is identically zero on the interval, the functions may be dependent (though this alone is not always conclusive). If the Wronskian is nonzero at even one point, the functions are guaranteed to be linearly independent. For solutions to a linear ODE, a vanishing Wronskian does confirm dependence.

Worked Example

Problem: Determine whether

f_1(x) = e^{2x}

and

f_2(x) = 5e^{2x}

are linearly dependent.

Step 1: Look for constants

c_1

and

c_2

, not both zero, such that

c_1 f_1 + c_2 f_2 = 0

for all

x

c_1 e^{2x} + c_2 (5e^{2x}) = 0

Step 2: Factor out

e^{2x}

, which is never zero.

e^{2x}(c_1 + 5c_2) = 0 \implies c_1 + 5c_2 = 0

Step 3: Choose

c_2 = 1

and

c_1 = -5

. These are not both zero, so a nontrivial combination exists.

-5e^{2x} + 1 \cdot 5e^{2x} = 0 \quad \checkmark

Answer: The functions are linearly dependent because

f_2(x) = 5f_1(x)

, so one is a constant multiple of the other.

Why It Matters

Identifying linearly dependent functions is essential when solving differential equations — you need linearly independent solutions to form a general solution. It also arises in signal processing when checking whether a set of basis functions is redundant.

Common Mistakes

Mistake: Assuming a Wronskian of zero at a single point proves dependence.

Correction: The Wronskian must vanish for all

x

in the interval to suggest dependence. A single zero value can occur even for independent functions (unless they are solutions to the same linear ODE).