Show that $f$ is a linear transformation

Question

Let $f: \mathbb{R}^m \to \mathbb{R}^n$, where $f(tx)=tf(x)$, for all $x \in \mathbb{R}^m$ and $t \in \mathbb{R}$. Show that $f$ is a linear transformation.

Answer 1

What you have provided is not enough to show that $ f $ is a linear transformation. It shows that $ f $ preserves scalar multiplication but does not show that $ f $ preserves addition. These two properties are independent of each other.

A non-trivial example is as follows. For each ray $ \ell $ through the origin in $ \mathbb{R}^{n} $, choose a scalar $ \alpha_{\ell} \in \mathbb{R} $. Then define $ f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ as follows: $$ \forall \mathbf{x} \in \mathbb{R}^{n}: \quad \mathbf{x} \in \ell \quad \Longrightarrow \quad f(\mathbf{x}) \stackrel{\text{def}}{=} \alpha_{\ell} \cdot \mathbf{x}. $$ It is seen that $ f $ scales each ray passing through the origin in a manner that depends only on the ray. Hence, by choosing the $ \alpha_{\ell} $'s carefully, one can make $ f $ non-linear.

Here is how we can do it. Pick $ \mathbf{x},\mathbf{y} \in \mathbb{R}_{n} \setminus \{ \mathbf{0} \} $ such that $ \mathbf{x} \notin \text{Span}(\{ \mathbf{y} \}) $. Let $ \ell(\mathbf{x}) $ and $ \ell(\mathbf{y}) $ be the unique rays passing through $ \mathbf{x} $ and $ \mathbf{y} $ respectively and also the origin. Clearly, $ \ell(\mathbf{x}) \neq \ell(\mathbf{y}) $. Set $ \alpha_{\ell(\mathbf{x})} = \alpha_{\ell(\mathbf{y})} = 1 $. Observe that $ \mathbf{x} + \mathbf{y} \neq \mathbf{0} $ and that $ \mathbf{x} + \mathbf{y} $ belongs to a unique ray $ \ell(\mathbf{x} + \mathbf{y}) $ passing through the origin that is identical to neither $ \ell(\mathbf{x}) $ nor $ \ell(\mathbf{y}) $. Now, if $ f $ were linear, we would expect to have \begin{align} f(\mathbf{x} + \mathbf{y}) &= f(\mathbf{x}) + f(\mathbf{y}) \\ &= \alpha_{\ell(\mathbf{x})} \cdot \mathbf{x} + \alpha_{\ell(\mathbf{y})} \cdot \mathbf{y} \\ &= 1 \cdot \mathbf{x} + 1 \cdot \mathbf{y} \\ &= \mathbf{x} + \mathbf{y} \\ &\neq \mathbf{0}. \end{align} However, as $ \ell(\mathbf{x} + \mathbf{y}) \neq \ell(\mathbf{x}),\ell(\mathbf{y}) $, we are free to choose $ \alpha_{\ell(\mathbf{x} + \mathbf{y})} = 0 $, which yields $ f(\mathbf{x} + \mathbf{y}) = \mathbf{0} $ instead. Therefore, $ f $ cannot be linear.

Answer 2

The correct definition is as follows:

Def: $f$ is linear if, as you write, $f(tx)=tf(x)$ for all vector $x$ and scalar $t$,

and $f(x+y)=f(x)+f(y)$ for all vectors $x,y$.

Your question as you stated is not true for $m>1$ (basically because it misses a part of the definition), and a counterexample is given in the comments by Daniel. Please show concrete example in $\Bbb R^2$ for $x,y$ such that $f(x+y)\ne f(x)+f(y)$ for that particular $f$.