Coordinate Transformations - Map from one system to another

Question

Let:

// T = transpose

e1 = [1 0]T
e2 = [0 1]T 

p = [2 2]T
a1 = [2 1]T
a2 = [-2 5]T

x = [2 1/2]T

be a point in the [e1 , e2]-system. In the new coordinate system, the [a1 , a2]-system, the coordinates of x define a new point xp.

What is this point with respect to the [e1 , e2]-system?

A = [
     2 -2
     1  4
]

xp = p + A * x;

// With respect to [e1, e2]
xp = [5 6]T

Thus, xp has coordinates [2 1/2]T with respect to [a1, a2]-system.

My Question

When we map x to [a1, a2]-system x remains the same for this system, but it changes from [e1, e2]-system's perspective?

Answer 1

Yes. x remains the same for [a1, a2]-system.