Give bases for row(A), col(A), and null(A). 10 - [1-3] 6 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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![**Title: Finding Bases for Row Space, Column Space, and Null Space**
**Objective:**
Determine the bases for the row space, column space, and null space of the matrix \( A \).
**Matrix:**
\[
A = \begin{bmatrix}
1 & 0 & -1 \\
1 & 1 & 6
\end{bmatrix}
\]
**Row Space, \( \text{row}(A) \):**
- The row space of a matrix consists of all possible linear combinations of its row vectors.
- Diagram: Two blank boxes to represent basis vectors, with green and grey arrows indicating relationships or dependencies among the components.
**Column Space, \( \text{col}(A) \):**
- The column space of a matrix consists of all possible linear combinations of its column vectors.
- Diagram: Two blank vertical boxes, with arrows suggesting the direction and linear combination within the columns.
**Null Space, \( \text{null}(A) \):**
- The null space of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector.
- Diagram: Two blank vertical sections, with arrows indicating dependencies that contribute to zero when combined with the matrix.
**Analysis:**
Understanding the relationship between these spaces:
- **Row space** reflects constraints imposed by equations derived from the rows.
- **Column space** considers spans generated by linear combinations of the original columns.
- **Null space** is crucial for solutions to homogeneous equations, indicating dependencies between variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1e36031-6f93-4211-8c57-892d2fd1b551%2Fb06ac607-d591-41d8-a71c-49c044378c6e%2Fd3fhqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding Bases for Row Space, Column Space, and Null Space**
**Objective:**
Determine the bases for the row space, column space, and null space of the matrix \( A \).
**Matrix:**
\[
A = \begin{bmatrix}
1 & 0 & -1 \\
1 & 1 & 6
\end{bmatrix}
\]
**Row Space, \( \text{row}(A) \):**
- The row space of a matrix consists of all possible linear combinations of its row vectors.
- Diagram: Two blank boxes to represent basis vectors, with green and grey arrows indicating relationships or dependencies among the components.
**Column Space, \( \text{col}(A) \):**
- The column space of a matrix consists of all possible linear combinations of its column vectors.
- Diagram: Two blank vertical boxes, with arrows suggesting the direction and linear combination within the columns.
**Null Space, \( \text{null}(A) \):**
- The null space of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector.
- Diagram: Two blank vertical sections, with arrows indicating dependencies that contribute to zero when combined with the matrix.
**Analysis:**
Understanding the relationship between these spaces:
- **Row space** reflects constraints imposed by equations derived from the rows.
- **Column space** considers spans generated by linear combinations of the original columns.
- **Null space** is crucial for solutions to homogeneous equations, indicating dependencies between variables.
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