Suppose is a linear transformation that is not one-to-one, and is not the trivial transformation, that is, for some v in V. Then .

Question

Suppose  is a linear transformation that is not one-to-one, and is not the trivial transformation, that is,  for some v in V. Then  .

Quizplus · Accepted Answer

Since the transformation is not one-to-one, there exists some u and v in V where u ≠ v but T(u) = T(v). Let w = u - v, then T(w) = T(u - v) = T(u) - T(v) = 0. Therefore, the transformation is not trivial. Then by the rank-nullity theorem, dim(ker(T)) > 0, which means there exists a non-zero vector in the kernel of the transformation.

Suppose Is a Linear Transformation That Is Not One-To-One