Real Function is Linearly Dependent with Zero Function
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Theorem
Let $\map f x$ be a real function defined on a closed interval $\closedint a b$.
Let $\map g x$ be the constant zero function on $\closedint a b$:
- $\forall x \in \closedint a b: \map g x = 0$
Then $f$ and $g$ are linearly dependent on $\closedint a b$.
Proof
We have that:
- $\forall x \in \closedint a b: \map g x = 0 = 0 \times \map f x$
and $0 \in \R$.
Hence the result by definition of linearly dependent real functions.
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.15$: The General Solution of the Homogeneous Equation