Answer
$$a = {2 \over 3},{\rm{ }}b = {1 \over 3}$$
Work Step by Step
$$\eqalign{
& {\rm{Let\ }}{\bf{u}} = \left\langle {1,2} \right\rangle ,{\rm{ }}{\bf{w}} = \left\langle {1, - 1} \right\rangle ,{\rm{ }}{\bf{v}} = \left\langle {1,1} \right\rangle \cr
& {\rm{Where\ }}{\bf{v}} = a{\bf{u}} + b{\bf{w}} \cr
& \left\langle {1,1} \right\rangle = a\left\langle {1,2} \right\rangle + b\left\langle {1, - 1} \right\rangle \cr
& \left\langle {1,1} \right\rangle = \left\langle {a,2a} \right\rangle + \left\langle {b, - b} \right\rangle \cr
& \left\langle {1,1} \right\rangle = \left\langle {a + b,2a - b} \right\rangle \cr
& {\rm{Therefore,}} \cr
& \left\{ \matrix{
a + b = 1 \hfill \cr
2a - b = 1 \hfill \cr} \right. \cr
& {\rm{Solving\ the\ system\ of\ equations}} \cr
& a + b = 1 \cr
& \underline {2a - b = 1} \cr
& 3a = 2 \cr
& a = {2 \over 3} \cr
& b = 1 - a = {1 \over 3} \cr
& {\rm{Then,}} \cr
& a = {2 \over 3},{\rm{ }}b = {1 \over 3} \cr} $$