8 Which vectors (b₁,b2, b3) are in the column space of A? Which combinations of the rows of A give zero? 1 (a) A = 2 263 0 2 5 (b) A = I 1 2 4 24 8
8 Which vectors (b₁,b2, b3) are in the column space of A? Which combinations of the rows of A give zero? 1 (a) A = 2 263 0 2 5 (b) A = I 1 2 4 24 8
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 8: Identifying Vectors in the Column Space and Row Combinations to Zero**
Given the matrices below:
**(a) Matrix \( A \)**
\[
A = \begin{bmatrix}
1 & 2 & 1 \\
2 & 6 & 3 \\
0 & 2 & 5
\end{bmatrix}
\]
**(b) Matrix \( A \)**
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 4 \\
2 & 4 & 8
\end{bmatrix}
\]
**Tasks:**
1. Determine which vectors \((b_1, b_2, b_3)\) are in the column space of \(A\).
2. Identify which combinations of the rows of \(A\) give zero.
**Explanation:**
- **Column Space**: The column space of a matrix \(A\) is the set of all possible linear combinations of its column vectors.
- **Row Combinations**: Finding row combinations that result in zero involves determining the linear dependence among the rows, often achieved through row reduction or examining the null space.
For each matrix:
**(a) Matrix \( A \) Analysis:**
- Column vectors:
\[
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix},
\begin{pmatrix}
2 \\
6 \\
2
\end{pmatrix},
\begin{pmatrix}
1 \\
3 \\
5
\end{pmatrix}
\]
- To find the vectors (b1, b2, b3) in the column space, solve the equation \(A \cdot x = b\) for consistency by column operations or row reduction.
- For row combinations giving zero, solve the homogenous system for rows.
**(b) Matrix \( A \) Analysis:**
- Column vectors:
\[
\begin{pmatrix}
1 \\
1 \\
2
\end{pmatrix},
\begin{pmatrix}
1 \\
2 \\
4
\end{pmatrix},
\begin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe243852a-5427-4fc7-bd56-ad30d22cf89a%2Fcb1637ba-4b79-437d-b993-4e4c916a15e0%2Fs7hozsv_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 8: Identifying Vectors in the Column Space and Row Combinations to Zero**
Given the matrices below:
**(a) Matrix \( A \)**
\[
A = \begin{bmatrix}
1 & 2 & 1 \\
2 & 6 & 3 \\
0 & 2 & 5
\end{bmatrix}
\]
**(b) Matrix \( A \)**
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 4 \\
2 & 4 & 8
\end{bmatrix}
\]
**Tasks:**
1. Determine which vectors \((b_1, b_2, b_3)\) are in the column space of \(A\).
2. Identify which combinations of the rows of \(A\) give zero.
**Explanation:**
- **Column Space**: The column space of a matrix \(A\) is the set of all possible linear combinations of its column vectors.
- **Row Combinations**: Finding row combinations that result in zero involves determining the linear dependence among the rows, often achieved through row reduction or examining the null space.
For each matrix:
**(a) Matrix \( A \) Analysis:**
- Column vectors:
\[
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix},
\begin{pmatrix}
2 \\
6 \\
2
\end{pmatrix},
\begin{pmatrix}
1 \\
3 \\
5
\end{pmatrix}
\]
- To find the vectors (b1, b2, b3) in the column space, solve the equation \(A \cdot x = b\) for consistency by column operations or row reduction.
- For row combinations giving zero, solve the homogenous system for rows.
**(b) Matrix \( A \) Analysis:**
- Column vectors:
\[
\begin{pmatrix}
1 \\
1 \\
2
\end{pmatrix},
\begin{pmatrix}
1 \\
2 \\
4
\end{pmatrix},
\begin
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