Real Function is Linearly Dependent with Zero Function

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Theorem

Let $f \left({x}\right)$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Let $g \left({x}\right)$ be the constant zero function on $\left[{a \,.\,.\, b}\right]$:

$\forall x \in \left[{a \,.\,.\, b}\right]: g \left({x}\right) = 0$


Then $f$ and $g$ are linearly dependent on $\left[{a \,.\,.\, b}\right]$.


Proof

We have that:

$\forall x \in \left[{a \,.\,.\, b}\right]: g \left({x}\right) = 0 = 0 \times f \left({x}\right)$

and $0 \in \R$.

Hence the result by definition of linearly dependent real functions.


Sources