Describe the possible echelon forms of the standard matrix for a linear transformation T where T: R³ R4 is one-to-one. Give some examples of the echelon forms. The leading entries, denoted, may have any nonzero value. The starred entries, denoted *, may have any value (including zero). Select all that apply. ☐A. 0 0 0 0 0 0 0 0 * * * * * B. [0 0 E. 0 0 0 0 0 0 0 * - 0 0 0 0 0 ▬ C. F. * 00 0 00 0 * 000 *
Describe the possible echelon forms of the standard matrix for a linear transformation T where T: R³ R4 is one-to-one. Give some examples of the echelon forms. The leading entries, denoted, may have any nonzero value. The starred entries, denoted *, may have any value (including zero). Select all that apply. ☐A. 0 0 0 0 0 0 0 0 * * * * * B. [0 0 E. 0 0 0 0 0 0 0 * - 0 0 0 0 0 ▬ C. F. * 00 0 00 0 * 000 *
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Possible Echelon Forms for a Linear Transformation
Describe the possible echelon forms of the standard matrix for a linear transformation \( T \) where \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^4 \) is one-to-one.
---
**Examples of Echelon Forms**
The leading entries, denoted \( \blacksquare \), may have any nonzero value. The starred entries, denoted \( \ast \), may have any value (including zero). Select all that apply.
- **Option A:**
\[
\begin{bmatrix}
\blacksquare & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
\end{bmatrix}
\]
- **Option B:**
\[
\begin{bmatrix}
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
0 & 0 & 0 & \blacksquare \\
\end{bmatrix}
\]
- **Option C:**
\[
\begin{bmatrix}
\ast & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
\end{bmatrix}
\]
- **Option D:**
\[
\begin{bmatrix}
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
0 & 0 & 0 & \ast \\
\end{bmatrix}
\]
- **Option E:**
\[
\begin{bmatrix}
\blacksquare & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & 0 & \blacksquare \\
\end{bmatrix}
\]
- **Option F:**
\[
\begin{bmatrix}
0 & \blacks](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2F7048d167-4c4e-438b-a329-dca6000afee6%2Fgo8ck8_processed.png&w=3840&q=75)
Transcribed Image Text:### Possible Echelon Forms for a Linear Transformation
Describe the possible echelon forms of the standard matrix for a linear transformation \( T \) where \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^4 \) is one-to-one.
---
**Examples of Echelon Forms**
The leading entries, denoted \( \blacksquare \), may have any nonzero value. The starred entries, denoted \( \ast \), may have any value (including zero). Select all that apply.
- **Option A:**
\[
\begin{bmatrix}
\blacksquare & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
\end{bmatrix}
\]
- **Option B:**
\[
\begin{bmatrix}
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
0 & 0 & 0 & \blacksquare \\
\end{bmatrix}
\]
- **Option C:**
\[
\begin{bmatrix}
\ast & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
\end{bmatrix}
\]
- **Option D:**
\[
\begin{bmatrix}
0 & \blacksquare & \ast & \ast \\
0 & 0 & \blacksquare & \ast \\
0 & 0 & 0 & \ast \\
\end{bmatrix}
\]
- **Option E:**
\[
\begin{bmatrix}
\blacksquare & \ast & \ast & \ast \\
0 & \blacksquare & \ast & \ast \\
0 & 0 & 0 & \blacksquare \\
\end{bmatrix}
\]
- **Option F:**
\[
\begin{bmatrix}
0 & \blacks
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