Question

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, with A its standard matrix. Complete the following statement to make it true: “T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if A has ____ pivot columns.” Find some theorems that explain why the statement is true.

Short answer

T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if A has \(m\) pivot columns.

Step-by-step solution

Identify the condition for the transformation of dimensions

For matrix Aof the order\(m \times n\), if the vector\({\bf{x}}\)is in\({\mathbb{R}^n}\), then transformation\(T\)of vector x is represented as\(T\left( x \right)\), and it is in the dimension\({\mathbb{R}^m}\).

It can also be written as\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\).

Here, the dimension \({\mathbb{R}^n}\) is the domain, and the dimension \({\mathbb{R}^m}\) is the codomain of transformation \(T\).

Complete the statement to make it true

By theorem, if the columns of A span (or generate)\({\mathbb{R}^m}\), then transformation\(T\)maps\({\mathbb{R}^n}\)onto\({\mathbb{R}^m}\).

If matrix Ahas a pivot position in each row, then\(T\)maps\({\mathbb{R}^n}\)onto\({\mathbb{R}^m}\).

The transformation\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)shows that there are\(m\)rows. So, it should have\(m\)pivot columns.

Thus, the correct statement is “T maps\({\mathbb{R}^n}\)onto\({\mathbb{R}^m}\)if and only if A has \(m\) pivot columns.”