Show that f is non linear and g is linear I have been given the Basis B of V and C of W in Q with
$$B = \{v_1,v_2,v_3\}$$
$$C = \{w_1,w_2\}$$
And f and g with:
$$f:\mathbb{V} \to \mathbb{W}, f(k_1v_1, k_2v_2, k_3v_3)=(3k_1+k_2)w_1+k_3^7w_2$$
$$g:\mathbb{V} \to \mathbb{W}, g(k_1v_1, k_2v_2, k_3v_3)=5k_1w_1+(2k_2+7k_3)w_2$$
I have to show that f is non linear and g is linear.
So I need to check for homogenity and additivity. So
$$f(kv) = kf(v)$$ and $$f(v+w) = f(v) + f(w)$$ has to be right for linearity. 
I see that f can't be linear because of the k^7.
But I have no clue how to put this Basis into the definition of linearity for this example.
I hope someone could show me.  
Greetings  

You're right about the $k_3^7$.  What's at work there is the freshman's binomial.  That is to say, were $(x+y)^7=x^7+y^7$ true, it would be linear.  But as we know, it isn't true.
