Matrix and vector problem

Question

Every vector  can be expressed uniquely in the form  where  is a scalar multiple of  and  is a scalar multiple of  Find the matrix  such that
for all vectors 

Answer 1

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 )