91Ó°ÊÓ

In example 1, the following matrices are in echelon form. The leading entries \(\left( \square \right)\) may have any nonzero value; the starred entries \(\left( * \right)\) may have any value (including zero).

\(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&\square & * & * & * & * & * & * & * & * \\ 0&0&0&\square & * & * & * & * & * & * \\ 0&0&0&0&\square & * & * & * & * & * \\ 0&0&0&0&0&\square & * & * & * & * \\ 0&0&0&0&0&0&0&0&\square & * \end{array}} \right]\)

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 \(T\)maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if the columns of \(A\) span\({\mathbb{R}^m}\).

Theorem 4states that let \(A\) be a \({\mathop{\rm m}\nolimits} \times n\) matrix,then \(A\) has a pivot position in every row.

The columns of \(A\) must span \({\mathbb{R}^3}\), according to theorem 12. The matrix contains a pivot in each row, according to theorem 4.

Use leading entries \(\left( \square \right)\) and starred entries \(\left( * \right)\) to write the possible echelon form of the standard matrix.

\(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&\square & * \end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&0&\square \end{array}} \right]\) ,

\(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&0&\square & * \\ 0&0&0&\square\end{array}} \right]\), and \(\left[ {\begin{array}{*{20}{c}}0&\square & * & * \\ 0&0&\square & * \\ 0&0&0&\square \end{array}} \right]\).

Therefore, \(T\) cannot be one-to-one because of the shape of \(A\).

Thus, the possible echelon form of the standard matrix is \(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&\square & * \end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&0&\square \end{array}} \right]\) , \(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&0&\square & * \\ 0&0&0&\square\end{array}} \right]\), and \(\left[ {\begin{array}{*{20}{c}}0&\square & * & * \\ 0&0&\square & * \\ 0&0&0&\square \end{array}} \right]\).