Linear transformation defined by T(a + bx) = (a, a+b)

[jeff1evesque]

Example
Let T: $$P_1(R) --> R^2$$ be the linear transformation defined by T(a + bx) = (a, a+b).
The reader can verify directly that T^-1: $$R^2 --> P_1(R)$$ is defined by T^-1(c, d) = c + (d-c)x. Observe that T^-1 is also linear.

I am reading my text and it kind of makes sense, but I have no clue how to verify what has been said above. It make sense to reverse everything, and because after the reversal since the inverse doesn't have the element c for the second element in the order pair for $$R^2$$ then we subtract it from the image of T^-1? But I don't feel comfortable with the concept (of this reversal-and subtraction), and I don't know how to verify what has been said.

 

[slider142]

If (c, d) is an element of the range of T (and thus the domain of T^-1), then c = a and d = a+b for some element (a + bx) in the domain of T. This should start you in the right direction.

 

[jeff1evesque]

If (c, d) is an element of the range of T (and thus the domain of T^-1), then c = a and d = a+b for some element (a + bx) in the domain of T. This should start you in the right direction.

Thank you,

JL