(a) Is this a subspace of R3? Justify your answer. (b) If W is a subspace of R³ then find the basis and the dimension of W.
(a) Is this a subspace of R3? Justify your answer. (b) If W is a subspace of R³ then find the basis and the dimension of W.
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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Transcribed Image Text:- (a) Is this a subspace of \( \mathbb{R}^3 \)? Justify your answer.
- (b) If \( W \) is a subspace of \( \mathbb{R}^3 \), then find the basis and the dimension of \( W \).
![Consider the set of all vectors
\[
W = \left\{
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix} \in \mathbb{R}^3 \text{ such that } 2x + 3y + z = 0
\right\}
\]
This expression defines a subset \( W \) within the three-dimensional real space \( \mathbb{R}^3 \). It consists of all vectors \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) where the components \( x \), \( y \), and \( z \) satisfy the linear equation \( 2x + 3y + z = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2Fe02649f1-0c7e-4fc1-96d4-1354d2fac59a%2Feree09a_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the set of all vectors
\[
W = \left\{
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix} \in \mathbb{R}^3 \text{ such that } 2x + 3y + z = 0
\right\}
\]
This expression defines a subset \( W \) within the three-dimensional real space \( \mathbb{R}^3 \). It consists of all vectors \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) where the components \( x \), \( y \), and \( z \) satisfy the linear equation \( 2x + 3y + z = 0 \).
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