Question

Why is the question “Is the linear transformation T onto?” an existence question?

Short answer

The given question is an existence question.

Step-by-step solution

Identify the condition for onto

If the range of transformation T includes the entire codomain \({\mathbb{R}^m}\), then T is onto \({\mathbb{R}^m}\). The transformation maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if at least one solution exists for \(T\left( {\bf{x}} \right) = {\bf{b}}\). Also, each vector b should be in the codomain \({\mathbb{R}^m}\).

Complete the statement to make it true

Consider vector u is in\({\mathbb{R}^m}\). All the vectors of u must be in the codomain\({\mathbb{R}^m}\), such that\(T\left( {\bf{x}} \right) = {\bf{u}}\). Also, in\({\mathbb{R}^n}\), vector x exists such that\(T\left( {\bf{x}} \right) = {\bf{u}}\), where\(T\left( {\bf{x}} \right) = {\bf{u}}\)is the linear transformation.

Thus, the given question is an existence question.