Let a=m(2,-1), b=n(3,1), so v=a+b=(2m+3n,-m+n).
Let P be a 2×2 matrix:
( x₁ y₁ )
( x₂ y₂ )
Then Pv=a can be represented by the equations:
x₁(2m+3n)+y₁(-m+n)=2m
x₂(2m+3n)+y₂(-m+n)=-m
From these we can get 4 equations by matching the coefficients of m and n.
(1) 2x₁-y₁=2
(2) 3x₁+y₁=0
(3) 2x₂-y₂=-1
(4) 3x₂+y₂=0
Add (1) and (2): 5x₁=2, x₁=⅖, so y₁=⅘-2=-⁶⁄₅.
Add (3) and (4): 5x₂=-1, x₂=-⅕, so y₂=-⅖+1=⅗.
Now we have P=
⅕( 2 -6 )
(-1 3 )