# 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?

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 $$\alpha \in \mathbb R.$$

Fred
July 12, 2019 09:41 AM