Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.CM: Cumulative Review
Problem 17CM: Find the sequence of the elementary matrices whose product is the non singular matrix below. [2410]
Related questions
Question
![**Problem Statement:**
Find bases for the row space, \( \text{row}(A) \), and the column space, \( \text{col}(A) \), in the given matrix using \( A^T \).
**Matrix Provided:**
\[
A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 8 \end{bmatrix}
\]
**Steps Illustrated:**
1. **Row Space \( \text{row}(A) \):**
- The row space consists of the linear combinations of the row vectors of \( A \).
- Typically, one performs row operations to row reduce the matrix to find independent rows.
2. **Column Space \( \text{col}(A) \):**
- The column space consists of the linear combinations of the column vectors of \( A \).
- Column operations or transposition can help in identifying independent columns.
**Diagrams and Arrows:**
- There are two main sets of diagrams:
- For \( \text{row}(A) \):
- Shows structured placement of vectors with arrows indicating operations to determine bases.
- For \( \text{col}(A) \):
- Indicates operations to distinguish independent columns through visual steps and highlighted arrows.
The diagrams use brackets to indicate the encapsulation of vectors with directional arrows to suggest focus areas for mathematical operations to identify the bases of the row and column spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6948d5d-782c-4f32-af22-abf47b4c3397%2F440a77bb-1192-443b-84a6-b71c5fb4f6ae%2Fcoap0c8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find bases for the row space, \( \text{row}(A) \), and the column space, \( \text{col}(A) \), in the given matrix using \( A^T \).
**Matrix Provided:**
\[
A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 8 \end{bmatrix}
\]
**Steps Illustrated:**
1. **Row Space \( \text{row}(A) \):**
- The row space consists of the linear combinations of the row vectors of \( A \).
- Typically, one performs row operations to row reduce the matrix to find independent rows.
2. **Column Space \( \text{col}(A) \):**
- The column space consists of the linear combinations of the column vectors of \( A \).
- Column operations or transposition can help in identifying independent columns.
**Diagrams and Arrows:**
- There are two main sets of diagrams:
- For \( \text{row}(A) \):
- Shows structured placement of vectors with arrows indicating operations to determine bases.
- For \( \text{col}(A) \):
- Indicates operations to distinguish independent columns through visual steps and highlighted arrows.
The diagrams use brackets to indicate the encapsulation of vectors with directional arrows to suggest focus areas for mathematical operations to identify the bases of the row and column spaces.
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