Let U=span(ū, ,ū,,ū and Û=span {u, ,u,, where ü,=(1,0,0,0), ū,=(0,1,1,0), ủ,=(0,1,1,1) and ů,=(1,0,0,1), ủ,=(1,1,0,0), ủ=(0,0,1,1). Show that U & Û are vector spaces (over the field R).
Let U=span(ū, ,ū,,ū and Û=span {u, ,u,, where ü,=(1,0,0,0), ū,=(0,1,1,0), ủ,=(0,1,1,1) and ů,=(1,0,0,1), ủ,=(1,1,0,0), ủ=(0,0,1,1). Show that U & Û are vector spaces (over the field R).
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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![**Transcription and Explanation:**
Let \( U = \text{span} \{\vec{u}_1, \vec{u}_2, \vec{u}_3\} \) and \( \hat{U} = \text{span} \{\vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3\} \) where
\[
\vec{u}_1 = (1, 0, 0, 0), \quad \vec{u}_2 = (0, 1, 1, 0), \quad \vec{u}_3 = (0, 1, 1, 1)
\]
and
\[
\vec{\hat{u}}_1 = (1, 0, 0, 1), \quad \vec{\hat{u}}_2 = (1, 1, 0, 0), \quad \vec{\hat{u}}_3 = (0, 0, 1, 1)
\]
Show that \( U \) and \( \hat{U} \) are vector spaces (over the field \( \mathbb{R} \)).
**Explanation:**
This text introduces two sets of vectors, \( \{\vec{u}_1, \vec{u}_2, \vec{u}_3\} \) and \( \{\vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3\} \), and asks to show that their spans, \( U \) and \( \hat{U} \), are vector spaces over the real numbers (\( \mathbb{R} \)).
- **\( \vec{u}_1, \vec{u}_2, \vec{u}_3 \):** These vectors are in a four-dimensional space and form the basis for the vector space \( U \).
- **\( \vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3 \):** These vectors are also in a four-dimensional space and form the basis for the vector space \( \hat{U} \).
To show that \( U \) and \( \hat{U} \) are vector](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a13d8c-5188-4010-ba23-710cc489cc25%2Fa2d3e207-0396-406f-9e4e-b280416c1f4e%2Fljb8v6l_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation:**
Let \( U = \text{span} \{\vec{u}_1, \vec{u}_2, \vec{u}_3\} \) and \( \hat{U} = \text{span} \{\vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3\} \) where
\[
\vec{u}_1 = (1, 0, 0, 0), \quad \vec{u}_2 = (0, 1, 1, 0), \quad \vec{u}_3 = (0, 1, 1, 1)
\]
and
\[
\vec{\hat{u}}_1 = (1, 0, 0, 1), \quad \vec{\hat{u}}_2 = (1, 1, 0, 0), \quad \vec{\hat{u}}_3 = (0, 0, 1, 1)
\]
Show that \( U \) and \( \hat{U} \) are vector spaces (over the field \( \mathbb{R} \)).
**Explanation:**
This text introduces two sets of vectors, \( \{\vec{u}_1, \vec{u}_2, \vec{u}_3\} \) and \( \{\vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3\} \), and asks to show that their spans, \( U \) and \( \hat{U} \), are vector spaces over the real numbers (\( \mathbb{R} \)).
- **\( \vec{u}_1, \vec{u}_2, \vec{u}_3 \):** These vectors are in a four-dimensional space and form the basis for the vector space \( U \).
- **\( \vec{\hat{u}}_1, \vec{\hat{u}}_2, \vec{\hat{u}}_3 \):** These vectors are also in a four-dimensional space and form the basis for the vector space \( \hat{U} \).
To show that \( U \) and \( \hat{U} \) are vector
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