Consider the following matrix A 1 2 1 3 2 Compute the rank of A. Compute the dimension of the nullspace of A (Hint: use the matrix rank theorem). Find a basis for the nullspace of A.
Consider the following matrix A 1 2 1 3 2 Compute the rank of A. Compute the dimension of the nullspace of A (Hint: use the matrix rank theorem). Find a basis for the nullspace of A.
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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Question
![**Linear Algebra Problem**
Consider the following matrix:
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
0 & 2 & 1 \\
1 & 3 & 2
\end{bmatrix}
\]
1. **Compute the rank of \( A \)**.
2. **Compute the dimension of the nullspace of \( A \)**.
*Hint: use the matrix rank theorem.*
3. **Find a basis for the nullspace of \( A \)**.
### Step-by-step Explanation:
1. **Rank of Matrix \( A \)**:
The rank of a matrix is the maximum number of linearly independent row vectors in the matrix. To find the rank:
- Perform row reduction to Row Echelon Form (REF).
- Count the number of non-zero rows.
2. **Dimension of the Nullspace**:
The dimension of the nullspace of \( A \), also known as the nullity, can be determined using the matrix rank theorem which states:
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
where \( n \) is the number of columns in the matrix \( A \).
3. **Basis for the Nullspace**:
To find a basis for the nullspace of \( A \):
- Solve the equation \( A \mathbf{x} = \mathbf{0} \).
- Express the solutions in terms of the free variables.
By carefully following these steps, one can compute the required rank, the dimension of the nullspace, and a basis for the nullspace for the given matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf4abcde-0790-4b6d-b8ff-52c0ea5b10aa%2F495271ee-0e9d-44b8-92f7-7c2962fccb8f%2Fr01o1io_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Problem**
Consider the following matrix:
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
0 & 2 & 1 \\
1 & 3 & 2
\end{bmatrix}
\]
1. **Compute the rank of \( A \)**.
2. **Compute the dimension of the nullspace of \( A \)**.
*Hint: use the matrix rank theorem.*
3. **Find a basis for the nullspace of \( A \)**.
### Step-by-step Explanation:
1. **Rank of Matrix \( A \)**:
The rank of a matrix is the maximum number of linearly independent row vectors in the matrix. To find the rank:
- Perform row reduction to Row Echelon Form (REF).
- Count the number of non-zero rows.
2. **Dimension of the Nullspace**:
The dimension of the nullspace of \( A \), also known as the nullity, can be determined using the matrix rank theorem which states:
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
where \( n \) is the number of columns in the matrix \( A \).
3. **Basis for the Nullspace**:
To find a basis for the nullspace of \( A \):
- Solve the equation \( A \mathbf{x} = \mathbf{0} \).
- Express the solutions in terms of the free variables.
By carefully following these steps, one can compute the required rank, the dimension of the nullspace, and a basis for the nullspace for the given matrix \( A \).
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