Question

Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?

Short answer

The linear transformation T cannot map \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).

Step-by-step solution

State the standard matrix of T

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and Abe the standard matrix for T. Then, according to theorem 12,

1. Tmaps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if the columns of Aspan \({\mathbb{R}^m}\);
2. T is one-to-one if and only if the columns of Aare linearly independent.

Consider Ais the standard matrix of T.

T is not one-to-one mapping according to the hypothesis. Therefore, according to theorem 12, the standard matrix A of Thas linearly dependent columns.

Determine whether T   maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)

The columns of A do not span \({\mathbb{R}^n}\) because Ais a square matrix. Therefore, the linear transformation T cannot map \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).