Give an example of a function 𝜑∶ 𝐑
2 → 𝐑 such that
𝜑(𝑎𝑣) = 𝑎𝜑(𝑣)
for all 𝑎 ∈ 𝐑 and all 𝑣 ∈ 𝐑2 but 𝜑 is not linear.

We are looking for a function \(\varphi: \mathbb{R}^2 \to \mathbb{R}\) that satisfies the condition:
\[
\varphi(av) = a\varphi(v) \quad \text{for all} \, a \in \mathbb{R} \, \text{and} \, v \in \mathbb{R}^2,
\]
but is **not linear**. A function is linear if it satisfies two properties:

1. **Additivity:** \(\varphi(v + w) = \varphi(v) + \varphi(w)\) for all \(v, w \in \mathbb{R}^2\).
2. **Homogeneity:** \(\varphi(av) = a \varphi(v)\) for all \(a \in \mathbb{R}\) and \(v \in \mathbb{R}^2\).

The condition we are given is homogeneity, but we want a function that does not satisfy additivity, so it will fail the linearity test.

### Example:

Consider the function \(\varphi: \mathbb{R}^2 \to \mathbb{R}\) defined by:
\[
\varphi(x, y) = \sqrt{x^2 + y^2}.
\]

#### Verification of Homogeneity:
For any scalar \(a \in \mathbb{R}\) and vector \(v = (x, y)\),
\[
\varphi(av) = \varphi(ax, ay) = \sqrt{(ax)^2 + (ay)^2} = \sqrt{a^2(x^2 + y^2)} = |a| \sqrt{x^2 + y^2} = a \varphi(x, y),
\]
because \(|a| = a\) when \(a \geq 0\) and \(|a| = -a\) when \(a < 0\). Hence, \(\varphi(av) = a\varphi(v)\) for all \(a \in \mathbb{R}\), satisfying the homogeneity condition.

#### Failing Additivity:
However, \(\varphi\) does not satisfy additivity. For example, if \(v = (1, 0)\) and \(w = (0, 1)\), we get:
\[
\varphi(v + w) = \varphi(1, 1) = \sqrt{1^2 + 1^2} = \sqrt{2},
\]
but
\[
\varphi(v) + \varphi(w) = \varphi(1, 0) + \varphi(0, 1) = 1 + 1 = 2.
\]
Since \(\sqrt{2} \neq 2\), \(\varphi\) does not satisfy additivity, and thus it is not linear.

### Conclusion:
The function \(\varphi(x, y) = \sqrt{x^2 + y^2}\) satisfies the homogeneity property but is not linear because it fails the additivity condition.

Would you like more details on this or have any questions?

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Here are 5 related questions to expand on this:

1. What are the general conditions for a function to be linear?
2. Can you find another function that satisfies homogeneity but is not linear?
3. How does additivity differ from homogeneity in linear algebra?
4. What are some geometric interpretations of homogeneity and linearity?
5. How does this example relate to norms and distances in vector spaces?

**Tip:** To check if a function is linear, always verify both additivity and homogeneity!

Give an example of a function 𝜑: 𝐑² → 𝐑 such that 𝜑(𝑎𝑣) = 𝑎𝜑(𝑣) for all 𝑎 ∈ 𝐑 and all 𝑣 ∈ 𝐑² but 𝜑 is not linear.

Explore a detailed example of a function 𝜑: 𝐑² → 𝐑 that satisfies the homogeneity property but fails to be linear due to a violation of additivity. This solution includes key concepts from linear algebra and offers a step-by-step breakdown. Ideal for students at the undergraduate level.

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