Find Coordinate Vector of A relatvie to {A_1, A_2, A_3, A_4, A_5, A_6}

[niteshadw]

How do you find the coordinate vector of

A =
1 1
1 2
2 2

relatvie to the basis {A_1, A_2, A_3, A_4, A_5, A_6}?

A_1 =
3 6
3 -6
0 0

A_2 =
0 -1
-1 0
1 1

A_3 =
0 -1
-2 -2
2 1

A_4 =
1 0
1 3
0 2

A_5 =
1 0
0 1
2 2

A_6 =
2 0
1 4
-1 3

Is it jus taking each of the matrix and making one large one and doing rref so I get:

88/157(A_1) + 23/157(A_2) + etc...?

I don't think its working out though...not even sure if its the correct thing to do...thanks

 

[HallsofIvy]

If I understand your notation correctly, you are looking for numbers x₁, x₂,..., x₆ so that
x₁A₁+ x₂A₂+ ...+ x₂A₂= A.

Looking at the individual components, that gives
3x₁+ x₄+ x₅+ 2x₆= 1,
6x₁- x₂- x₃= 1, etc.
6 equations in 6 unknowns. Of course, you can set that up as a matrix equation itself.