How to prove that a function is linear with vector

Question

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

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

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

Thanks!

Answer 1

I think that $ ()=<,(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 $\alpha \in \mathbb R.$