Is T invertible?

Question

Let T:R^3➜R^2 be defined as T(x, y, z) = (x+y, y-z).

Is T Invertible?

How can you justify your answer?

Answer 1

Here are some steps to work through:

1) If T is a transformation from R^3 to R^2, what are its dimensions?

Hint: multiplying T by a 3x1 matrix (vector) results in a 2x1 matrix (vector).

2) Is this an nxn square matrix? Is a non-square matrix invertible?

3) Think about your answer conceptually. It makes sense to collapse a vector from R^3 to R^2, but you can't reverse the transformation and expand a vector that exists in R^2 into R^3 - you can't bring a 3rd dimension into being where there previously was none.