Question: In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation \(T\). Use the notation of Example 1 in section 1.2.

29. \(T:{\mathbb{R}^3} \to {\mathbb{R}^4}\) is one-to-one.

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:

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.

The columns of \(A\) must be linearly independent, and so the equation \(Ax = 0\) contains no free variable according to theorem 12. Therefore, \(A\) must have a pivot column in each column.

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

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

The shape of \(A\) prevents \(T\) from being one-to-one.

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