Is this linear transformation one-to-one?

[The Student]

Let A be a 3 by 2 matrix. Can the transformation x

Ax be one-to-one? My answer is "yes" if the bottom row of A are zeros. I can't see why not because if the span of A is R2 and the span of x is R2, then it seems as though Ax could be R2. But I am never shocked anymore when I'm wrong.

 

[HallsofIvy]

Your example is correct. As long as the rank of the matrix is 2, it will map \(\displaystyle R^2\) one to one onto a plane in \(\displaystyle R^3\).

 

[JohnZ]

As long as the rank of the matrix is 2, it will map R2 one to one onto a plane in R3.

it's one to one but not onto

 

[The Student]

Thanks!!!