Let W be the set of all vectors C2 C3 CA such that 1-2x2 = 423 and 2x1 = 3 + 3x4 Determine if W is a vector space and check the correct answer(s) below. A. W is a vector space because it can be written as N(A) for some matrix A. B. W is not a vector space because it does not have a zero element.
Let W be the set of all vectors C2 C3 CA such that 1-2x2 = 423 and 2x1 = 3 + 3x4 Determine if W is a vector space and check the correct answer(s) below. A. W is a vector space because it can be written as N(A) for some matrix A. B. W is not a vector space because it does not have a zero element.
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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![### Vector Spaces in Linear Algebra
Let \( W \) be the set of all vectors
\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
\]
such that \( x_1 - 2x_2 = 4x_3 \) and \( 2x_1 = x_3 + 3x_4 \).
### Determine if \( W \) is a vector space and check the correct answer(s) below.
#### Options:
- **A.** \( W \) is a vector space because it can be written as \( N(A) \) for some matrix \( A \).
- **B.** \( W \) is not a vector space because it does not have a zero element.
- **C.** \( W \) is not a vector space because it is not closed with respect to scalar multiplication.
- **D.** \( W \) is a vector space because it is in \( \mathbb{R}^4 \).
- **E.** \( W \) is a vector space because it has a zero element.
- **F.** \( W \) is not a vector space because it does not have additive closure.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F63f9830d-c769-4f0a-a8af-e3fa3e6acbc3%2Fk0x984z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector Spaces in Linear Algebra
Let \( W \) be the set of all vectors
\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
\]
such that \( x_1 - 2x_2 = 4x_3 \) and \( 2x_1 = x_3 + 3x_4 \).
### Determine if \( W \) is a vector space and check the correct answer(s) below.
#### Options:
- **A.** \( W \) is a vector space because it can be written as \( N(A) \) for some matrix \( A \).
- **B.** \( W \) is not a vector space because it does not have a zero element.
- **C.** \( W \) is not a vector space because it is not closed with respect to scalar multiplication.
- **D.** \( W \) is a vector space because it is in \( \mathbb{R}^4 \).
- **E.** \( W \) is a vector space because it has a zero element.
- **F.** \( W \) is not a vector space because it does not have additive closure.
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