Please show work U=span{ū,ū,,ū, andÛ=span {u ,u,,ú} whereū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) andů,=(1,0,0,1), u=(1,1,0,0), u=(0,0,1,1).2. Let%3D%3DShow thatU & Ûare vector spaces (overthe field R).

(a) Given vectors are u1^→=1,0,0,1,u2^→=1,1,0,0,u3^→=0,0,1,1. Consider U^=spanu1^→, u2^→, u3^→.…

Answered: Let U=span{uq ,U2,U3} Û=span {u,…

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

Please show work

U=span{ū,ū,,ū, and
Û=span {u ,u,,ú} where
ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and
ů,=(1,0,0,1), u=(1,1,0,0), u=(0,0,1,1).
2. Let
%3D
%3D
Show that
U & Û
are vector spaces (over
the field R).

Expert Solution

Step 1

(a)

Given vectors are $\vec{\hat{u_{1}}} = (1, 0, 0, 1), \vec{\hat{u_{2}}} = (1, 1, 0, 0), \vec{\hat{u_{3}}} = (0, 0, 1, 1)$ .

Consider $\hat{U} = s p a n \{\vec{\hat{u_{1}}}, \vec{\hat{u_{2}}}, \vec{\hat{u_{3}}}\}$ .

Clearly the set $\hat{U} = s p a n \{\vec{\hat{u_{1}}}, \vec{\hat{u_{2}}}, \vec{\hat{u_{3}}}\}$ is non empty as three given vectors are already there. Since the set

$\hat{U} = s p a n \{\vec{\hat{u_{1}}}, \vec{\hat{u_{2}}}, \vec{\hat{u_{3}}}\}$ is already subset of a vector space $ℝ^{4}$ , it is enough to show that the set

$\hat{U} = s p a n \{\vec{\hat{u_{1}}}, \vec{\hat{u_{2}}}, \vec{\hat{u_{3}}}\}$ is closed under vector addition and scalar multiplication.

Let $\vec{\hat{x}}, \hat{\vec{y}} \in \hat{U}$ be arbitrary vectors. Therefore they can be written as:

$\begin{array}{rcl} \vec{\hat{x}} & = & a \vec{\hat{u_{1}}} + b \vec{\hat{u_{2}}} + c \vec{\hat{u_{3}}} \\ \vec{\hat{y}} & = & l \vec{\hat{u_{1}}} + m \vec{\hat{u_{2}}} + n \vec{\hat{u_{3}}} \end{array}$

Therefore there vector addition:

$\begin{array}{rcl} \vec{\hat{x}} + \vec{\hat{y}} & = & (a \vec{\hat{u_{1}}} + b \vec{\hat{u_{2}}} + c \vec{\hat{u_{3}}}) + (l \vec{\hat{u_{1}}} + m \vec{\hat{u_{2}}} + n \vec{\hat{u_{3}}}) \\ = & (a + l) \vec{\hat{u_{1}}} + (b + m) \vec{\hat{u_{2}}} + (c + n) \vec{\hat{u_{3}}} \\ \in & \hat{U} \end{array}$

Hence the set $\hat{U} = s p a n \{\vec{\hat{u_{1}}}, \vec{\hat{u_{2}}}, \vec{\hat{u_{3}}}\}$ is closed under vector addition.

Let $k \in ℝ$ be an arbitrary scalar. Therefore there scalar multiplication can be written as:

$\begin{array}{rcl} k \vec{\hat{x}} & = & k (a \vec{\hat{u_{1}}} + b \vec{\hat{u_{2}}} + c \vec{\hat{u_{3}}}) \\ = & (k a) \vec{\hat{u_{1}}} + (k b) \vec{\hat{u_{2}}} + (k c) \vec{\hat{u_{3}}} \\ \in & \hat{U} \end{array}$