Let U=span u, ,ū2, ūz and Ü=span{ú, ,ú2,úz where ü,=(1,0,0,0), ủ,=(0,1,1,0), ủ,=(0,1,1,1) and u,=(1,0,0,1), ủ,=(1,1,0,0), u,=(0,0,1,1). Show that UnÛ ± Ø, i.e. given some üeU, describe üeÛ in terms of üeU, explicitly.
Let U=span u, ,ū2, ūz and Ü=span{ú, ,ú2,úz where ü,=(1,0,0,0), ủ,=(0,1,1,0), ủ,=(0,1,1,1) and u,=(1,0,0,1), ủ,=(1,1,0,0), u,=(0,0,1,1). Show that UnÛ ± Ø, i.e. given some üeU, describe üeÛ in terms of üeU, explicitly.
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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![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 \( U \cap \hat{U} \neq \emptyset \), i.e., given some \( \vec{u} \in U \), describe \( \vec{\hat{u}} \in \hat{U} \) in terms of \( \vec{u} \), explicitly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a13d8c-5188-4010-ba23-710cc489cc25%2F00f292e9-3836-4994-b2e3-7e3012eb4b51%2Fqw8m3f_processed.png&w=3840&q=75)
Transcribed Image Text: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 \( U \cap \hat{U} \neq \emptyset \), i.e., given some \( \vec{u} \in U \), describe \( \vec{\hat{u}} \in \hat{U} \) in terms of \( \vec{u} \), explicitly.
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