Determine which of the following sets is a vector space. V is the line y - x in the xy-plane: V - {{} -} W is the union of the first and second quadrants in the xy-plane: W U is the line y = x+ 1 in the xy-plane: U = A) U only B) V only C) w only D) U and V
Determine which of the following sets is a vector space. V is the line y - x in the xy-plane: V - {{} -} W is the union of the first and second quadrants in the xy-plane: W U is the line y = x+ 1 in the xy-plane: U = A) U only B) V only C) w only D) U and V
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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![### Determining Vector Spaces
#### Question 2:
**Determine which of the following sets is a vector space.**
1. **Set V**: V is the line \( y = x \) in the xy-plane.
\[ V = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y = x \right\} \]
2. **Set W**: W is the union of the first and second quadrants in the xy-plane.
\[ W = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y \ge 0 \right\} \]
3. **Set U**: U is the line \( y = x + 1 \) in the xy-plane.
\[ U = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y = x + 1 \right\} \]
**Options:**
A) U only
B) V only
C) W only
D) U and V
**Explanation:**
- **Set V** (the line \( y = x \)): This set is a vector space because it satisfies the conditions for vector spaces, including closure under addition and scalar multiplication, contains the zero vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \), and is not empty.
- **Set W** (the union of the first and second quadrants \( y \ge 0 \)): This set is not a vector space because it does not include all scalar multiples of its vectors. For example, \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \in W \) but \( -1 \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \end{pmatrix} \notin W \).
- **Set U** (the line \( y = x + 1 \)): This set is not a vector space because it does not include the zero vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and does not satisfy closure under addition and scalar multiplication.
**Correct Answer:**
B) V only](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2Fe8cfaaed-d01d-402d-a815-a0f5c2dc62f8%2F5fhdfz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining Vector Spaces
#### Question 2:
**Determine which of the following sets is a vector space.**
1. **Set V**: V is the line \( y = x \) in the xy-plane.
\[ V = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y = x \right\} \]
2. **Set W**: W is the union of the first and second quadrants in the xy-plane.
\[ W = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y \ge 0 \right\} \]
3. **Set U**: U is the line \( y = x + 1 \) in the xy-plane.
\[ U = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : y = x + 1 \right\} \]
**Options:**
A) U only
B) V only
C) W only
D) U and V
**Explanation:**
- **Set V** (the line \( y = x \)): This set is a vector space because it satisfies the conditions for vector spaces, including closure under addition and scalar multiplication, contains the zero vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \), and is not empty.
- **Set W** (the union of the first and second quadrants \( y \ge 0 \)): This set is not a vector space because it does not include all scalar multiples of its vectors. For example, \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \in W \) but \( -1 \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \end{pmatrix} \notin W \).
- **Set U** (the line \( y = x + 1 \)): This set is not a vector space because it does not include the zero vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and does not satisfy closure under addition and scalar multiplication.
**Correct Answer:**
B) V only
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