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B112045005 吳愷杰 302-2--d,e,f

##### Exercise 2(d)

判斷 $f: \mathbb{R}^2\rightarrow\mathbb{R}$ 且 $f(x,y) = x^2 + y^2$ 是否線性。

<!-- eng start -->
Let $f: \mathbb{R}^2\rightarrow\mathbb{R}$ be a function defined by $f(x,y) = x^2 + y^2$. Is it linear?
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:::warning

- [x] we got --> we get
- [x] Thus --> Thus,
:::

##### Exercise 2(d) -- answer here:

Because $f(kx,ky) = k^2x^2 + k^2y^2$ and $kf(x,y) = kx^2 + ky^2$, we get $kf(x,y) \neq f(kx,ky)$，
Thus, $f$ is not linear.

##### Exercise 2(e)

判斷 $f: \mathbb{R}^2\rightarrow\mathbb{R}$ 且 $f(x,y) = 3x + 2y$ 是否線性。

<!-- eng start -->
Let $f: \mathbb{R}^2\rightarrow\mathbb{R}$ be a function defined by $f(x,y) = 3x + 2y$. Is it linear?
<!-- eng end -->

:::warning

- [x] It should be a period after $f(x_2, y_2)$.
- [x] that's why --> so
:::

##### Exercise 2(e) -- answer here

Because $kf(x,y) = 3kx + 2ky = f(kx,ky)$ and
$$\begin{align}
f(x_1 + x_2,y_1 + y_2) & = 3(x_1 + x_2) + 2(y_1 + y_2) \\
& = 3x_1 + 3x_2 + 2y_1 + 2y_2 \\
& = f(x_1,y_1) + f(x_2,y_2).
\end{align}
$$
It fulfills the definition, so $f$ is linear.

:::warning

- [x] Because ..., $f$ is not linear.
:::

##### Exercise 2(f)

判斷 $f: \mathbb{R}^2\rightarrow\mathbb{R}$ 且 $f(x,y) = 3x + 2y + 1$ 是否線性。

<!-- eng start -->
Let $f: \mathbb{R}^2\rightarrow\mathbb{R}$ be a function defined by $f(x,y) = 3x + 2y + 1$. Is it linear?
<!-- eng end -->

##### Exercise 2(f) -- answer here:

Beacuse $kf(x,y) = 3kx + 2ky + k \neq f(kx,ky)$, $f$ is not linear.