Indirect proof: Let be singular and its twin transformation. Since is singular, there exists an with . Thus we have , and we also have that , a contradiction.

Since is non-singular we have that for every basis vector there exists a uniquely determined , such that .2. Prove that if T nonsingular, so there is only one "twin transformations".

Therefore the component of the image of the basis vector under in the direction of is given by , and thus is uniquely determined by .