Answer
See explanation
Work Step by Step
Given
\[
T: \mathbb{R}^{4} \mapsto \mathbb{R}^{3} \text { is onto }
\]
Goal
Describe the possible echelon shapes of the standard matrix for a linear transformation $T$.
Concepts
Definition of Onto Mapping
Plan:
1. Determine the size of the matrix for the given mapping.
2. Determine the possible echelon shapes of the standard matrix.From the given transformation, we know that the size of standard matrix $A$ is $3 \times 4 .$ since the possible echelon forms of $A$ must be onto in this case, we must make all possible shapes $A$ such that its columns span $\mathbb{R}^{m}$. (Recall Theorem 4). The columns of $A$ will span $\mathbb{R}^{m}$ if $A$ has a pivot position in every row. Knowing this, the possible echelon shapes of the standard matrix are
\[
\begin{array}{l}
A_{1}=\left[\begin{array}{llll}
\mathbf{I} & * & * & * \\
0 & \mathbf{I} & * & * \\
0 & 0 & \mathbf{I} & *
\end{array}\right] \\
A_{2}=\left[\begin{array}{llll}
0 & \mathbf{l} & * & * \\
0 & 0 & \mathbf{g} & * \\
0 & 0 & 0 & \mathbf{D}
\end{array}\right] \\
A_{3}=\left[\begin{array}{llll}
\mathbf{\square} & * & * & * \\
0 & \mathbf{D} & * & * \\
0 & 0 & 0 & \mathbf{D}
\end{array}\right] \\
A_{4}=\left[\begin{array}{llll}
\mathbf{D} & * & * & * \\
0 & 0 & \mathbf{D} & * \\
0 & 0 & 0 & \mathbf{D}
\end{array}\right]
\end{array}
\]
Conclusion:
The possible echelon shapes of the standard matrix
