How to prove that a function is linear with vector

by root   Last Updated July 12, 2019 10:20 AM - source

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?


Answers 1

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 $\alpha \in \mathbb R.$

July 12, 2019 09:41 AM

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