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 is one-to-one if and only if A has ­­____ pivot columns.” Explain why the statement is true. [Hint:Look in the exercises for Section 1.7.]

Short answer

T is one-to-one if and only if A has ­­\(n\) pivot columns.

Step-by-step solution

Identify the condition for the transformation of dimensions

For matrix Aof 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

If the columns of A are linearly independent, then transformation T is one-to-one.

Matrix A has n pivot columns if they are linearly independent.

Thus, the correct statement is “T is one-to-one if and only if A has ­­\(n\) pivot columns.”