Find both a basis for the row space and a basis for the column space of the given matrix A. 14 3 15-2 3 8 29 A basis for the row space is . (Use a comma to separate matrices as needed.) A basis for the column space is (Use a comma to separate matrices as needed.)
Find both a basis for the row space and a basis for the column space of the given matrix A. 14 3 15-2 3 8 29 A basis for the row space is . (Use a comma to separate matrices as needed.) A basis for the column space is (Use a comma to separate matrices as needed.)
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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![### Finding a Basis for Row Space and Column Space
Given the matrix \( A \):
\[ \begin{bmatrix}
1 & 4 & 3 \\
1 & 5 & -2 \\
3 & 8 & 29
\end{bmatrix} \]
We want to determine:
1. A basis for the row space.
2. A basis for the column space.
**Instructions**:
1. A **basis** for the row space is [input needed].
(Use a comma to separate matrices as needed.)
2. A **basis** for the column space is [input needed].
(Use a comma to separate matrices as needed.)
---
### Detailed Explanation
#### Row Space:
The row space of matrix \( A \) is the space spanned by its rows. To find a basis for the row space, we perform row reduction to echelon form:
\[ \begin{bmatrix}
1 & 4 & 3 \\
1 & 5 & -2 \\
3 & 8 & 29
\end{bmatrix} \rightarrow
\begin{bmatrix}
1 & 4 & 3 \\
0 & 1 & -5 \\
0 & 0 & 1
\end{bmatrix} \]
The non-zero rows in the echelon form matrix span the row space.
#### Column Space:
The column space of matrix \( A \) is the space spanned by its columns. To find a basis for the column space, we look at the original columns of \( A \).
Since we perform row operations in echelon form, the pivotal columns indicate the columns needed for a basis. These columns in the original matrix \( A \) form the basis.
#### Example:
For this matrix, assuming row reduction has been completed correctly:
- Row space basis might look like: (specific vectors to be input)
- Column space basis might look like: (specific columns to be input)
Please calculate the echelon form and pivotal columns for exact answers.
---
This guide will help you fill out the blank matrices in the provided boxes, ensuring you identify the correct bases for both row and column spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a1a8c69-ac20-486b-9f67-8f66504c5494%2F2136c5dd-155b-42da-996c-335de0115108%2Fi90fwbv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding a Basis for Row Space and Column Space
Given the matrix \( A \):
\[ \begin{bmatrix}
1 & 4 & 3 \\
1 & 5 & -2 \\
3 & 8 & 29
\end{bmatrix} \]
We want to determine:
1. A basis for the row space.
2. A basis for the column space.
**Instructions**:
1. A **basis** for the row space is [input needed].
(Use a comma to separate matrices as needed.)
2. A **basis** for the column space is [input needed].
(Use a comma to separate matrices as needed.)
---
### Detailed Explanation
#### Row Space:
The row space of matrix \( A \) is the space spanned by its rows. To find a basis for the row space, we perform row reduction to echelon form:
\[ \begin{bmatrix}
1 & 4 & 3 \\
1 & 5 & -2 \\
3 & 8 & 29
\end{bmatrix} \rightarrow
\begin{bmatrix}
1 & 4 & 3 \\
0 & 1 & -5 \\
0 & 0 & 1
\end{bmatrix} \]
The non-zero rows in the echelon form matrix span the row space.
#### Column Space:
The column space of matrix \( A \) is the space spanned by its columns. To find a basis for the column space, we look at the original columns of \( A \).
Since we perform row operations in echelon form, the pivotal columns indicate the columns needed for a basis. These columns in the original matrix \( A \) form the basis.
#### Example:
For this matrix, assuming row reduction has been completed correctly:
- Row space basis might look like: (specific vectors to be input)
- Column space basis might look like: (specific columns to be input)
Please calculate the echelon form and pivotal columns for exact answers.
---
This guide will help you fill out the blank matrices in the provided boxes, ensuring you identify the correct bases for both row and column spaces.
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