Let V = M₂x2 and let W = Span The vector True False The vector and let W = Span{ [3] [85] []} True False [32] [² 0 Let V = M2₂x2 and let W = Span [24] [13] [1 is in W. 1 11 2 -10 -4 344] is in W.
Let V = M₂x2 and let W = Span The vector True False The vector and let W = Span{ [3] [85] []} True False [32] [² 0 Let V = M2₂x2 and let W = Span [24] [13] [1 is in W. 1 11 2 -10 -4 344] is in 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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Question
![### Linear Algebra: Span and Vector Spaces
In this exercise, you will determine whether a given vector is inside the span of a set of matrices.
---
**Problem 1:**
Let \( V = M_{2 \times 2} \) and let
\[ W = \text{Span}\left\{ \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 5 \\ 0 & -3 \end{pmatrix}, \begin{pmatrix} 1 & -2 \\ 0 & 2 \end{pmatrix} \right\}. \]
The vector
\[ \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix} \]
is in \( W \).
- True
- False
---
**Problem 2:**
Let \( V = M_{2 \times 2} \) and let
\[ W = \text{Span}\left\{ \begin{pmatrix} 2 & 1 \\ -2 & -3 \end{pmatrix}, \begin{pmatrix} 1 & -2 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 4 \\ 4 & -1 \end{pmatrix} \right\}. \]
The vector
\[ \begin{pmatrix} 1 & 2 \\ -10 & -4 \end{pmatrix} \]
is in \( W \).
- True
- False
---
For each problem, select either "True" or "False" based on whether the given vector can be written as a linear combination of the matrices in the span \( W \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F510b87f1-1062-4180-bcf7-c90378c42705%2F21f636b0-67a9-454d-95e2-93e0acefbf9a%2F08qfxob_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Span and Vector Spaces
In this exercise, you will determine whether a given vector is inside the span of a set of matrices.
---
**Problem 1:**
Let \( V = M_{2 \times 2} \) and let
\[ W = \text{Span}\left\{ \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 5 \\ 0 & -3 \end{pmatrix}, \begin{pmatrix} 1 & -2 \\ 0 & 2 \end{pmatrix} \right\}. \]
The vector
\[ \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix} \]
is in \( W \).
- True
- False
---
**Problem 2:**
Let \( V = M_{2 \times 2} \) and let
\[ W = \text{Span}\left\{ \begin{pmatrix} 2 & 1 \\ -2 & -3 \end{pmatrix}, \begin{pmatrix} 1 & -2 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 4 \\ 4 & -1 \end{pmatrix} \right\}. \]
The vector
\[ \begin{pmatrix} 1 & 2 \\ -10 & -4 \end{pmatrix} \]
is in \( W \).
- True
- False
---
For each problem, select either "True" or "False" based on whether the given vector can be written as a linear combination of the matrices in the span \( W \).
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