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  

this just means that you have (for B) three linearly independent vectors that span your vector space V. It is a more general way of writing vecor spaces, you don't always have a standard basis (i.e. $\mathbb{R}^{2}=span{((1,0), (0,1))}$).
For your example you can simply handle this the exact way you would if you had been given a standard basis, i.e.:
$f$ is not linear:
Let $v\in V, k\in K$, then
$$f(k_{1}v_{1}+l_{1}v_1,k_{2}v_2+l_2v_2, k_{3}v_3+l_3v_3) = 3((k_1+l_1+k_2+l_2) w_1 + (k_3+l_3)^7w_2
$$
And continue on from there.
