Determine if the columns of the matrix form a linearly independent set. Justify your answer. -2 -1 0 -1 1 1 - 14 2 1 -28 Select the correct choice below and fill in the answer box within your choice. Type an integer or simplified fraction for each matrix element.) O A. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A do not form linearly independent set. O B. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax =0 has only the trivial solution. Therefore, the columns of A form a linearly independent set. O C. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax =0 has only the trivial solution. Therefore, the columns of A do not form a linearly independent set. O D. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A form a linearly independent set.
Determine if the columns of the matrix form a linearly independent set. Justify your answer. -2 -1 0 -1 1 1 - 14 2 1 -28 Select the correct choice below and fill in the answer box within your choice. Type an integer or simplified fraction for each matrix element.) O A. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A do not form linearly independent set. O B. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax =0 has only the trivial solution. Therefore, the columns of A form a linearly independent set. O C. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax =0 has only the trivial solution. Therefore, the columns of A do not form a linearly independent set. O D. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A form a linearly independent set.
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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![**Title: Analyzing the Linear Independence of Matrix Columns**
**Objective:** Determine if the columns of the given matrix form a linearly independent set and justify your answer.
**Matrix Representation:**
\[
\begin{bmatrix}
-2 & -1 & 0 \\
0 & -1 & 7 \\
1 & 1 & -14 \\
2 & 1 & -28
\end{bmatrix}
\]
**Question:** Examine the columns of the matrix and assess whether they constitute a linearly independent set by evaluating the solutions to the equation \( Ax = 0 \).
**Options for Consideration:**
- **Option A:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has more than one solution.
- Therefore, the columns of \( A \) do not form a linearly independent set.
- **Option B:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has only the trivial solution.
- Therefore, the columns of \( A \) form a linearly independent set.
- **Option C:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has only the trivial solution.
- Therefore, the columns of \( A \) do not form a linearly independent set.
- **Option D:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has more than one solution.
- Therefore, the columns of \( A \) form a linearly independent set.
**Instructions:** Select the correct option and fill in the blank in the answer box with your choice, ensuring](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d31c5b4-c6dc-4739-be04-6ff777f4b60f%2F9855e618-4f32-47a4-94fa-51e312faf280%2F979gmjf_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing the Linear Independence of Matrix Columns**
**Objective:** Determine if the columns of the given matrix form a linearly independent set and justify your answer.
**Matrix Representation:**
\[
\begin{bmatrix}
-2 & -1 & 0 \\
0 & -1 & 7 \\
1 & 1 & -14 \\
2 & 1 & -28
\end{bmatrix}
\]
**Question:** Examine the columns of the matrix and assess whether they constitute a linearly independent set by evaluating the solutions to the equation \( Ax = 0 \).
**Options for Consideration:**
- **Option A:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has more than one solution.
- Therefore, the columns of \( A \) do not form a linearly independent set.
- **Option B:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has only the trivial solution.
- Therefore, the columns of \( A \) form a linearly independent set.
- **Option C:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has only the trivial solution.
- Therefore, the columns of \( A \) do not form a linearly independent set.
- **Option D:**
- If \( A \) is the given matrix, then the augmented matrix \(\boxed{\text{ }}\) represents the equation \( Ax = 0 \).
- The reduced echelon form of this matrix indicates that \( Ax = 0 \) has more than one solution.
- Therefore, the columns of \( A \) form a linearly independent set.
**Instructions:** Select the correct option and fill in the blank in the answer box with your choice, ensuring
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