Question

In Exercises 25-28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer.

27. The transformation in Exercise 19.

Short answer

The specified linear transformation is onto.

Step-by-step solution

The transformation in Exercise 19

Write the transformation in Exercise 19.

\(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1} - 5{x_2} + 4{x_3},{x_2} - 6{x_3}} \right)\)

Determine the standard matrix of \(T\) by inspection

Theorem 12states that let\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)be a linear transformation, and let\(A\)be the standard matrix\(T\)then:

1. \(T\)maps\({\mathbb{R}^n}\)onto\({\mathbb{R}^m}\)if and only if the columns of\(A\)span\({\mathbb{R}^m}\).
2. \(T\)is one-to-one if and only if the columns of\(A\)are linearly independent.

Matrix \(A\) has linearly independent columns since the figure indicates that \({a_1}\) and \({a_2}\) are not multiples. Thus, \(T\) is one-to-one, according to theorem 12.

Determine whether the linear transformation is one-to-one or onto

Theorem 12states that let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and let \(A\) be the standard matrix \(T\) then:

1. \(T\)maps \(({\mathbb{R}^n}\))) onto \(({\mathbb{R}^m}\))) if and only if the columns of \(A\) span \(({\mathbb{R}^m}\))).
2. \(T\)is one-to-one if and only if the columns of \(A\) are linearly independent.

Matrix \(A\) has more columns than rows; so the columns of \(A\) are linearly dependent. Thus, \(T\) is not one-to-one, according to theorem 12. Moreover, the column of \(A\) has a pivot in each row; so the rows of \(A\) spans \({\mathbb{R}^2}\). Therefore,\(T\) maps \({\mathbb{R}^3}\), according to theorem 12.

Thus, the specified linear transformation is onto.