[Math] Zero Function Linear

linear algebralinear-transformations

Is $f: \mathbb{R^m} \rightarrow \mathbb{R^n}$ defined by $f(\vec x) \equiv 0 $ considered a linear map? If the definition for linear map is one that satisfies additivity and homogeneity, then $f$ obviously is a linear map. But I'd to confirm $f$ is not specifically excluded in the usual definitions of linear mappings.

Best Answer

Yes, the zero function is indeed a linear map.

Any function that is additive and homogeneous is a linear map by definition, and in your case:

  • $f(x+y) = 0 = 0+0 = f(x) + f(y)$
  • $f(\alpha x) = 0 = \alpha \cdot 0 = \alpha f(x)$

So $f$ is both homogeneous and additive.

In fact, as one commenter pointed out, you can define addition and multiplication with constant to the set of all linear functions.

  • $(f+g)(x) = f(x) + g(x)$
  • $(\alpha\cdot f)(x) = \alpha \cdot (f(x))$

These two definitions are well defined and map linear functions to linear functions.

Using these two operations, the set of all linear functions from $\mathbb R^m$ to $\mathbb R^n$ is a linear space (which, by the way, is isomorphic to $\mathbb R^{m\cdot n}$) and is more commonly viewed as the space of all $n\times m$ matrices) and the zero function is the zero element.

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