Write the vectors u, v, w, z in terms of a and b. 
Write the vectors $u, v, w, z$ in terms of $a$ and $b$.


I'm unsure of how to do this.. If someone could give me an example of one being done I'm almost positive I could mimic it and figure out the rest
 A: Simply draw multiples of $a$'s and $b$'s so that you get a path to the other vectors, in this way you'll have their combination. I think this is a purely drawing-geometrical exercise. For example if you add $b$ on $a$, you get $u$, as you move one left and two above. Adding $a$ to $a$ and then $b$ to this, you get $w$ and so on.

*

*$a$ = one right, one above (call it) $(1,1)$

*$b$ = one left, two above (call it) $(-1,2)$
so we have $2*a + b = (2-1,2+2)=(1,4)=w$
Remember that for some of them you'll have to add negative $a$'s or negative $b$'s, which you do by simply writing the vector in the same direction but opposite sense.
For example we have $ b +(-a) = v $
A: You need to get the points $u,v,w,z$ in relation of the coordinates and then get a linear combination of $a,b$ such that is equal to them. $u=(0,3)$, $v=(-2,1)$, $w=(1,4)$, $z=(3,0)$. Here is an example:
$$
2a-b=2(1,1)-(-1,2)=(2,2)+(1,-2)=(3,0)=z
$$
The other ones are here in case you want to try it before check an answer

 $$ -a+b=-(1,1)+(-1,2)=(-1,-1)+(-1,2)=(-2,1)=v $$
 $$ 2a+b=2(1,1)+(-1,2)=(2,2)+(-1,2)=(1,4)=w $$
 $$ a+b=(1,1)+(-1,2)=(1,1)+(-1,+2)=(0,3)=u $$

