10-6 03-5 and b -59 4 a. Is b in (a, a, a)? How many vectors are in (a,. a. az)? b. Is b in W? How many vectors are in W? c. Show that az is in W. (Hint: Row operations are unnecessary.] Let A Denote the columns of A by a₁. 2. 3. and let W-Span (₁, 2, 3). a. Is b in (a,, a2. a)? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. No, b is not in (a,. a2. a), since b is not equal to a,, a2, or ag. B. Yes, b is in (a,. a2. a) since ba (Type a whole number.) C. Yes, b is in (a,. a2. a3), since, although b is not equal to a,. a2. or ay, it can be expressed as a linear combination of them. In particular, b= (a₁ + a₂ + (ªz. (Simplify your answers.) D. No, b is not in (a, a, a) since it cannot be generated by a linear combination of a,, a2, and a.
10-6 03-5 and b -59 4 a. Is b in (a, a, a)? How many vectors are in (a,. a. az)? b. Is b in W? How many vectors are in W? c. Show that az is in W. (Hint: Row operations are unnecessary.] Let A Denote the columns of A by a₁. 2. 3. and let W-Span (₁, 2, 3). a. Is b in (a,, a2. a)? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. No, b is not in (a,. a2. a), since b is not equal to a,, a2, or ag. B. Yes, b is in (a,. a2. a) since ba (Type a whole number.) C. Yes, b is in (a,. a2. a3), since, although b is not equal to a,. a2. or ay, it can be expressed as a linear combination of them. In particular, b= (a₁ + a₂ + (ªz. (Simplify your answers.) D. No, b is not in (a, a, a) since it cannot be generated by a linear combination of a,, a2, 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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![On this educational page, we will explore some concepts in linear algebra, specifically in the context of vector spaces and linear combinations.
---
**Given:**
Matrix \( A \) and vector \( b \)
\[ A = \begin{bmatrix}
1 & 0 & -6 \\
0 & 3 & -5 \\
-5 & 9 & 4
\end{bmatrix}
\quad \text{and} \quad
b = \begin{bmatrix}
9 \\
-2 \\
-29
\end{bmatrix} \]
**Columns of \( A \)**
Let's denote the columns of \( A \) by \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \). Define \( W \) as the span of these columns:
\[ W = \text{Span} \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} \]
**Questions:**
1. Is \( b \) in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)? How many vectors are in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)?
2. Is \( b \) in \( W \)? How many vectors are in \( W \)?
3. Show that \( \mathbf{a}_2 \) is in \( W \). [Hint: Row operations are unnecessary.]
---
### Detailed Problem Solving
a. **Is \( b \) in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)?**
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \( \mathbf{A.} \) No, \( b \) is not in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\) since \( b \) is not equal to \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \).
- \( \mathbf{B.} \) Yes, \( b \) is in \(\{ \mathbf{a}_1, \mathbf{a}_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c3522a4-50b1-4726-a83e-d332776e45d2%2Fb23d2861-9613-426d-ac0f-a3e0280e06b7%2F0ik28bk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this educational page, we will explore some concepts in linear algebra, specifically in the context of vector spaces and linear combinations.
---
**Given:**
Matrix \( A \) and vector \( b \)
\[ A = \begin{bmatrix}
1 & 0 & -6 \\
0 & 3 & -5 \\
-5 & 9 & 4
\end{bmatrix}
\quad \text{and} \quad
b = \begin{bmatrix}
9 \\
-2 \\
-29
\end{bmatrix} \]
**Columns of \( A \)**
Let's denote the columns of \( A \) by \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \). Define \( W \) as the span of these columns:
\[ W = \text{Span} \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} \]
**Questions:**
1. Is \( b \) in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)? How many vectors are in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)?
2. Is \( b \) in \( W \)? How many vectors are in \( W \)?
3. Show that \( \mathbf{a}_2 \) is in \( W \). [Hint: Row operations are unnecessary.]
---
### Detailed Problem Solving
a. **Is \( b \) in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\)?**
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \( \mathbf{A.} \) No, \( b \) is not in \(\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}\) since \( b \) is not equal to \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \).
- \( \mathbf{B.} \) Yes, \( b \) is in \(\{ \mathbf{a}_1, \mathbf{a}_
![### Linear Combinations of Vectors
Use the accompanying figure to write each vector listed as a linear combination of \( u \) and \( v \).
**Vectors:**
- \( b \)
- \( w \)
- \( x \)
- \( y \)
#### Explanation of Diagram:
The diagram is a Cartesian plane with vectors \( u \) and \( v \) depicted along with their multiples and combinations. The vectors of interest (\( b \), \( w \), \( x \), and \( y \)) are marked on the grid. Each marked point on the grid corresponds to a particular linear combination of the basis vectors \( u \) and \( v \).
Here are the markers of different vectors:
- \(O\) represents the origin (0,0).
- \(u\) and \(v\) represent the unit vectors in the directions of \(u\) and \(v\), respectively.
- Other points are arranged in a grid based on multiples and sums of these basis vectors.
#### Write Vector **b** as a Linear Combination
Write \( b \) as a linear combination of \( u \) and \( v \):
\[ b = (\underline{\hspace{1cm}}) u + (\underline{\hspace{1cm}}) v \]
>(*Type integers or decimals.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c3522a4-50b1-4726-a83e-d332776e45d2%2Fb23d2861-9613-426d-ac0f-a3e0280e06b7%2Fykrjk0a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Combinations of Vectors
Use the accompanying figure to write each vector listed as a linear combination of \( u \) and \( v \).
**Vectors:**
- \( b \)
- \( w \)
- \( x \)
- \( y \)
#### Explanation of Diagram:
The diagram is a Cartesian plane with vectors \( u \) and \( v \) depicted along with their multiples and combinations. The vectors of interest (\( b \), \( w \), \( x \), and \( y \)) are marked on the grid. Each marked point on the grid corresponds to a particular linear combination of the basis vectors \( u \) and \( v \).
Here are the markers of different vectors:
- \(O\) represents the origin (0,0).
- \(u\) and \(v\) represent the unit vectors in the directions of \(u\) and \(v\), respectively.
- Other points are arranged in a grid based on multiples and sums of these basis vectors.
#### Write Vector **b** as a Linear Combination
Write \( b \) as a linear combination of \( u \) and \( v \):
\[ b = (\underline{\hspace{1cm}}) u + (\underline{\hspace{1cm}}) v \]
>(*Type integers or decimals.)*
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