I have to show that the function 𝑓(𝑥)=<𝑥,(34)> is a linear function.

I understand that the proof that is not linear 𝑓(𝑥+𝑦)≠𝑓(𝑥)+𝑓(𝑦).

But honestly I have no idea where to start to prove it. Any ideas or advice?

Thanks!

I think that $ 𝑓(𝑥)=<𝑥,(34)>$ has the following meaning: $(3,4) $ is a given vector in $\mathbb R^2$ and with $x=(x_1,x_2) \in \mathbb R^2$ we have

$$f(x)=<(x_1,x_2),(3,4)>,$$

where $< \cdot,\cdot>$ denotes the usual inner product on $ \mathbb R^2.$ Hence

$$f(x)=3x_1+4x_2.$$

Now it is your turn to show that

$$f(x+y)=f(x)+f(y)$$

and

$$f( \alpha x)=\alpha f(x)$$

for all $x,y \in \mathbb R^2$ and all $\alpha \in \mathbb R.$

July 12, 2019 09:41 AM

- Serverfault Help
- Superuser Help
- Ubuntu Help
- Webapps Help
- Webmasters Help
- Programmers Help
- Dba Help
- Drupal Help
- Wordpress Help
- Magento Help
- Joomla Help
- Android Help
- Apple Help
- Game Help
- Gaming Help
- Blender Help
- Ux Help
- Cooking Help
- Photo Help
- Stats Help
- Math Help
- Diy Help
- Gis Help
- Tex Help
- Meta Help
- Electronics Help
- Stackoverflow Help
- Bitcoin Help
- Ethereum Help