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I'm having trouble understanding the purpose of using two basis in a linear transformation. My lecturer explained that it was a way to find a linear transformation that satisfied either dimension, but I'm having trouble understanding how that relates to the method in finding this transformation.

An example being,

T(Ao+A

_{1}x+A

_{2}x

^{2})=(Ao+A

_{2})+Aox

Using standard basis

b = (Ao,A

_{1}x,A

_{2}x

^{2})

C= (1,x)

[T]cb =

| 1 0 1 |

| 1 0 0 |, what does this indicate?