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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!

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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 $\alpha \in \mathbb R.$

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