I understand that, for example, if we have a 2 by 2 matrix, when its determinant is zero, then it does not have an inverse matrix. Hence, the linear transformation is non-invertible.
If we have a 1 by 1 square on the 2D plane, then, geometrically, this non-invertible 2D matrix transforms it into a 1D line segment.
However, on the other hand, don't a line segment and a square have the same amount of points? If so, there has to be a mapping that maps each point on this segment to the square, right? So does this mean even though the matrix is non-invertible, the transformation itself is still invertible?
Thank you for your time answering!