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Recipe

Prove a set of functions is linearly independent

Use when: you have a finite list of functions and must show no nontrivial linear combination of them equals the zero function

Goal: a chain of forced equations c_1=c_2=\dots=0, proving the only combination equal to zero is the trivial one

Taught in Infinite-dimensional spaces: polynomials, sequences, functions

Quick recipe

  1. Write the defining equation
  2. Choose a strategy that isolates one coefficient at a time
  3. Apply the chosen strategy, one coefficient at a time
  4. Confirm every coefficient is forced to zero
  1. 1

    Set c_1f_1(x)+\dots+c_nf_n(x)=0, required to hold for every x in the domain, not just one chosen value.

    Example

    For \{\sin x,\cos x\}: c_1\sin x + c_2\cos x = 0 for every x.

    Why: independence means this equation forces every coefficient to zero, so writing it precisely is the whole target of the proof.

  2. 2

    Either substitute well-chosen x-values, or differentiate before substituting, whichever eliminates every term but one.

    • the functions are monomials like 1,x,x^2,\dots — differentiate repeatedly, since each derivative permanently kills the lower powers.
    • the functions are transcendental, like \sin x and \cos x — evaluate directly at points where all but one function vanishes.

    Why: each choice is just a tool for turning one equation in n unknowns into n separate one-variable equations.

  3. 3

    Carry out the substitutions or differentiations in order, solving for each coefficient as it becomes isolated.

    Example

    At x=0: c_1(0)+c_2(1)=0 forces c_2=0. At x=\pi/2: c_1(1)+c_2(0)=0 forces c_1=0.

    Why: each substitution is chosen precisely so every term except one drops out, leaving a one-variable equation to solve immediately.

  4. 4

    Check that the chain of substitutions or derivatives covered every coefficient in the original list, not just some of them.

    Example

    Both c_1 and c_2 came out zero, covering the whole list \{\sin x,\cos x\}, with nothing left unchecked.

    Why: independence requires ALL coefficients to be zero, so a proof pinning down only some of them isn't finished.

Common mistakes

  • Testing only one x-value, or one derivative, and assuming that pins down every coefficient.
  • Forgetting the defining equation must hold for every x, not just the specific values chosen to test it.
  • A sign or arithmetic slip evaluating \sin or \cos at a standard angle.
  • Stopping once some coefficients are forced to zero without checking every coefficient in the list.