Let -3 ----------- V3 2 V1 = 6 a. How many vectors are in {V₁, V2, V3}? b. How many vectors are in Col A? c. Is D in Col A? Why or why not? 11 and A
Let -3 ----------- V3 2 V1 = 6 a. How many vectors are in {V₁, V2, V3}? b. How many vectors are in Col A? c. Is D in Col A? Why or why not? 11 and 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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![### Problem 7:
Let
\[ \mathbf{v}_1 = \begin{bmatrix} 2 \\ -8 \\ 6 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -3 \\ 8 \\ -7 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -4 \\ 6 \\ -7 \end{bmatrix}, \mathbf{p} = \begin{bmatrix} 6 \\ -10 \\ 11 \end{bmatrix}, \text{ and } A = [\mathbf{v}_1 \mathbf{v}_2 \mathbf{v}_3]. \]
a. How many vectors are in \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} \)?
b. How many vectors are in Col A?
c. Is \( \mathbf{p} \) in Col A? Why or why not?
---
- **Vector Notation Explanation:**
- \(\mathbf{v}_1\), \(\mathbf{v}_2\), \(\mathbf{v}_3\) are vectors in \(\mathbb{R}^3\), each represented as a column matrix.
- \(\mathbf{p}\) is also a vector in \(\mathbb{R}^3\).
- \(A\) is a matrix formed by the column vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\).
- **Graphs/Diagrams Explanation:**
- There are no graphs or diagrams provided in this problem statement. The problem consists solely of vector and matrix notation and related questions based on these.
---
### Solutions Guide:
a. In the set \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} \), there are 3 vectors.
b. The number of vectors in Col A is equivalent to the number of columns in matrix \(A\). Therefore, there are 3 vectors in Col A.
c. To determine whether \( \mathbf{p} \) is in Col A, we need to check if \( \mathbf{p} \) can be expressed as a linear combination of the vectors \( \mathbf{v}_1, \mathbf{v](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12a9a3e9-d100-4e80-b765-c2c582e29977%2F507c564b-c927-4be5-8803-535cdae40810%2Fh8tzzee_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 7:
Let
\[ \mathbf{v}_1 = \begin{bmatrix} 2 \\ -8 \\ 6 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -3 \\ 8 \\ -7 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -4 \\ 6 \\ -7 \end{bmatrix}, \mathbf{p} = \begin{bmatrix} 6 \\ -10 \\ 11 \end{bmatrix}, \text{ and } A = [\mathbf{v}_1 \mathbf{v}_2 \mathbf{v}_3]. \]
a. How many vectors are in \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} \)?
b. How many vectors are in Col A?
c. Is \( \mathbf{p} \) in Col A? Why or why not?
---
- **Vector Notation Explanation:**
- \(\mathbf{v}_1\), \(\mathbf{v}_2\), \(\mathbf{v}_3\) are vectors in \(\mathbb{R}^3\), each represented as a column matrix.
- \(\mathbf{p}\) is also a vector in \(\mathbb{R}^3\).
- \(A\) is a matrix formed by the column vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\).
- **Graphs/Diagrams Explanation:**
- There are no graphs or diagrams provided in this problem statement. The problem consists solely of vector and matrix notation and related questions based on these.
---
### Solutions Guide:
a. In the set \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} \), there are 3 vectors.
b. The number of vectors in Col A is equivalent to the number of columns in matrix \(A\). Therefore, there are 3 vectors in Col A.
c. To determine whether \( \mathbf{p} \) is in Col A, we need to check if \( \mathbf{p} \) can be expressed as a linear combination of the vectors \( \mathbf{v}_1, \mathbf{v
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