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?



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


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


Now it is your turn to show that



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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.